WEBVTT 00:00:00.000 --> 00:00:01.960 align:middle line:90% [SQUEAKING] 00:00:01.960 --> 00:00:03.920 align:middle line:90% [RUSTLING] 00:00:03.920 --> 00:00:07.350 align:middle line:90% [CLICKING] 00:00:07.350 --> 00:00:19.130 align:middle line:90% 00:00:19.130 --> 00:00:30.270 align:middle line:84% CASEY RODRIGUEZ: OK, so the main character today, 00:00:30.270 --> 00:00:36.580 align:middle line:84% in this lecture and the next, are compact operators, 00:00:36.580 --> 00:00:40.800 align:middle line:84% which I introduced at the end of last lecture. 00:00:40.800 --> 00:00:57.230 align:middle line:84% And I'll recall throughout, H is a Hilbert space. 00:00:57.230 --> 00:01:01.740 align:middle line:90% 00:01:01.740 --> 00:01:11.130 align:middle line:84% So we say an operator, let's call it K, 00:01:11.130 --> 00:01:20.080 align:middle line:84% is a bounded linear operator, is a compact operator, 00:01:20.080 --> 00:01:31.590 align:middle line:84% if the image of the unit ball, taking the closure of that-- 00:01:31.590 --> 00:01:38.060 align:middle line:84% so this is just a set of all elements K u with u 00:01:38.060 --> 00:01:53.140 align:middle line:84% less than or equal to 1 closure is compact in H. 00:01:53.140 --> 00:01:58.750 align:middle line:84% And how did this set of operators come up? 00:01:58.750 --> 00:02:00.880 align:middle line:84% At the end of last time, we dealt 00:02:00.880 --> 00:02:03.430 align:middle line:84% with finite rank operators, which 00:02:03.430 --> 00:02:04.930 align:middle line:90% were just basically matrices. 00:02:04.930 --> 00:02:08.740 align:middle line:84% You can actually write them so that they're completely 00:02:08.740 --> 00:02:13.360 align:middle line:84% described with respect to a certain basis as a matrix. 00:02:13.360 --> 00:02:15.880 align:middle line:84% And the question we posed at the end of the last lecture 00:02:15.880 --> 00:02:19.930 align:middle line:84% was, these finite rank operators, 00:02:19.930 --> 00:02:21.610 align:middle line:84% they form a subspace of the space 00:02:21.610 --> 00:02:23.080 align:middle line:90% of all bounded linear operators. 00:02:23.080 --> 00:02:24.310 align:middle line:90% Is that subspace closed? 00:02:24.310 --> 00:02:25.600 align:middle line:90% And the answer is no. 00:02:25.600 --> 00:02:30.370 align:middle line:84% What is the closure of the finite rank 00:02:30.370 --> 00:02:31.810 align:middle line:84% operators, which we know and love 00:02:31.810 --> 00:02:33.610 align:middle line:90% because they're just matrices? 00:02:33.610 --> 00:02:35.950 align:middle line:84% And what we'll prove today is that it's exactly 00:02:35.950 --> 00:02:38.440 align:middle line:90% the space of compact operators. 00:02:38.440 --> 00:02:40.930 align:middle line:90% But let me just-- 00:02:40.930 --> 00:02:46.510 align:middle line:84% last time, I gave you an example of one 00:02:46.510 --> 00:02:53.330 align:middle line:84% that is a compact operator K a equal to, 00:02:53.330 --> 00:02:58.640 align:middle line:84% let's say, a1 divided by a1 one, a2 divided by 2, 00:02:58.640 --> 00:03:06.110 align:middle line:84% a3 divided by 3, and so on, defined for a in l2, little l2. 00:03:06.110 --> 00:03:13.440 align:middle line:84% Then my claim is that K is a compact operator. 00:03:13.440 --> 00:03:16.440 align:middle line:84% And in the assignment, you show basically a variety 00:03:16.440 --> 00:03:19.050 align:middle line:84% of other operators that come up quite often 00:03:19.050 --> 00:03:20.310 align:middle line:90% are compact operators. 00:03:20.310 --> 00:03:26.340 align:middle line:84% For example, you also show, essentially, 00:03:26.340 --> 00:03:39.270 align:middle line:84% that if K is a continuous function on 0, 1, cross 0, 1, 00:03:39.270 --> 00:03:43.350 align:middle line:84% and I define a linear operator T f 00:03:43.350 --> 00:03:48.750 align:middle line:84% of x to be the integral from 0 to 1 of K x, y, 00:03:48.750 --> 00:03:56.730 align:middle line:84% f of y dy, now f is in big L2 of 0, 1. 00:03:56.730 --> 00:04:01.680 align:middle line:84% Then part of the assignment is to use and remind yourself 00:04:01.680 --> 00:04:04.110 align:middle line:90% of one of the only-- 00:04:04.110 --> 00:04:05.490 align:middle line:90% one of the very few-- 00:04:05.490 --> 00:04:08.160 align:middle line:84% compactness results one has, is to use 00:04:08.160 --> 00:04:10.750 align:middle line:84% the Arzela-Ascoli theorem to show that this is, 00:04:10.750 --> 00:04:12.790 align:middle line:90% in fact, a compact operator. 00:04:12.790 --> 00:04:15.360 align:middle line:84% So from the assignment, you'll prove 00:04:15.360 --> 00:04:24.790 align:middle line:90% T is compact operator on L2. 00:04:24.790 --> 00:04:27.800 align:middle line:90% 00:04:27.800 --> 00:04:31.640 align:middle line:84% Now, maybe you think I'm just coming up 00:04:31.640 --> 00:04:33.140 align:middle line:84% with something that looks like this 00:04:33.140 --> 00:04:35.930 align:middle line:84% to come up with examples of a compact operator, 00:04:35.930 --> 00:04:42.590 align:middle line:84% but the solution operator for differential equations 00:04:42.590 --> 00:04:44.630 align:middle line:90% take this form. 00:04:44.630 --> 00:04:47.420 align:middle line:90% So for example-- 00:04:47.420 --> 00:04:49.170 align:middle line:84% I guess this would be an example, 00:04:49.170 --> 00:04:50.940 align:middle line:90% as part of this example-- 00:04:50.940 --> 00:04:55.340 align:middle line:90% if I take K x, y to be-- 00:04:55.340 --> 00:04:57.920 align:middle line:84% let's see if I can get it right-- 00:04:57.920 --> 00:05:06.700 align:middle line:84% x minus 1 times y, x times y minus 1, 00:05:06.700 --> 00:05:11.140 align:middle line:84% for 0 less than or equal to y less than 00:05:11.140 --> 00:05:13.030 align:middle line:84% or equal to x less than or equal to 1. 00:05:13.030 --> 00:05:17.195 align:middle line:90% And then now reverse. 00:05:17.195 --> 00:05:22.190 align:middle line:90% 00:05:22.190 --> 00:05:22.750 align:middle line:90% So let's see. 00:05:22.750 --> 00:05:25.270 align:middle line:90% 00:05:25.270 --> 00:05:27.060 align:middle line:90% Yeah, I think that's it. 00:05:27.060 --> 00:05:29.720 align:middle line:90% 00:05:29.720 --> 00:05:35.120 align:middle line:90% Then u equals 0, 1 K of x, y. 00:05:35.120 --> 00:05:36.290 align:middle line:90% You can actually compute. 00:05:36.290 --> 00:05:41.460 align:middle line:90% 00:05:41.460 --> 00:05:44.490 align:middle line:84% Solves the differentiable equation 00:05:44.490 --> 00:05:56.250 align:middle line:84% u double prime equals f, u of 0 equals u of 1 equals 0 on 0, 1. 00:05:56.250 --> 00:06:01.650 align:middle line:84% So maybe this has a minus in front. 00:06:01.650 --> 00:06:03.030 align:middle line:90% I can't exactly remember-- 00:06:03.030 --> 00:06:07.170 align:middle line:90% 00:06:07.170 --> 00:06:09.930 align:middle line:90% u of x. 00:06:09.930 --> 00:06:16.320 align:middle line:84% So the solution operator, meaning if my input is f, 00:06:16.320 --> 00:06:18.900 align:middle line:84% then the solution to this differential equation 00:06:18.900 --> 00:06:19.740 align:middle line:90% is actually-- 00:06:19.740 --> 00:06:21.570 align:middle line:84% I can write it as a compact operator 00:06:21.570 --> 00:06:23.670 align:middle line:90% acting on the data, which is f. 00:06:23.670 --> 00:06:28.760 align:middle line:90% 00:06:28.760 --> 00:06:31.310 align:middle line:84% So these don't just pop up out of nowhere. 00:06:31.310 --> 00:06:35.240 align:middle line:90% They pop up quite naturally. 00:06:35.240 --> 00:06:41.240 align:middle line:84% Now, these are a bunch of examples on compact operators. 00:06:41.240 --> 00:06:45.280 align:middle line:84% One non-example-- it's easy to actually come up 00:06:45.280 --> 00:06:47.740 align:middle line:84% with non-examples of compact operators-- 00:06:47.740 --> 00:06:54.430 align:middle line:84% but let's say the identity on little l2. 00:06:54.430 --> 00:06:56.200 align:middle line:90% This is not a compact operator. 00:06:56.200 --> 00:07:03.040 align:middle line:90% 00:07:03.040 --> 00:07:03.790 align:middle line:90% Why? 00:07:03.790 --> 00:07:08.860 align:middle line:84% Because the condition of being a compact operator 00:07:08.860 --> 00:07:12.670 align:middle line:84% is that K of the unit ball should be 00:07:12.670 --> 00:07:16.570 align:middle line:90% a compact subset of little l2. 00:07:16.570 --> 00:07:24.180 align:middle line:84% But let's let e sub n be the sequence-- 00:07:24.180 --> 00:07:26.400 align:middle line:84% our favorite sequence-- which is 0, 00:07:26.400 --> 00:07:28.440 align:middle line:84% except in the n-th spot, where it's 1. 00:07:28.440 --> 00:07:34.280 align:middle line:90% 00:07:34.280 --> 00:07:39.220 align:middle line:90% Then each of these has norm 1. 00:07:39.220 --> 00:07:42.310 align:middle line:90% 00:07:42.310 --> 00:07:51.450 align:middle line:84% And so if I look at the image, e n, that's I e m, 00:07:51.450 --> 00:07:53.340 align:middle line:84% and I compute the square of that, 00:07:53.340 --> 00:07:56.970 align:middle line:84% you can see quite easily this is equal to 2 00:07:56.970 --> 00:08:02.940 align:middle line:84% since these are orthonormal for all n not equal to m. 00:08:02.940 --> 00:08:19.530 align:middle line:84% And therefore, I e n does not have a convergent subsequence. 00:08:19.530 --> 00:08:22.530 align:middle line:84% If I, the identity, was a compact operator, then 00:08:22.530 --> 00:08:26.610 align:middle line:84% the image of the closed unit ball in little l2 00:08:26.610 --> 00:08:29.130 align:middle line:84% should be a compact set, meaning what? 00:08:29.130 --> 00:08:34.710 align:middle line:84% Meaning elements of this form, I evaluated 00:08:34.710 --> 00:08:37.409 align:middle line:84% on something that has norm less than or equal to 1, 00:08:37.409 --> 00:08:41.650 align:middle line:90% should form a compact set. 00:08:41.650 --> 00:08:44.440 align:middle line:84% So if I have a sequence of guys in the unit ball, 00:08:44.440 --> 00:08:46.570 align:middle line:84% closed unit ball, then when I hits it, 00:08:46.570 --> 00:08:51.590 align:middle line:84% I should be able to find a convergent subsequence. 00:08:51.590 --> 00:08:54.410 align:middle line:84% But here, if you just compute, then we 00:08:54.410 --> 00:08:59.800 align:middle line:84% have the distance between any two of these guys for n 00:08:59.800 --> 00:09:00.880 align:middle line:90% not equal to m is 2. 00:09:00.880 --> 00:09:02.950 align:middle line:84% So this sequence can't ever be Cauchy, 00:09:02.950 --> 00:09:05.440 align:middle line:84% nor can any subsequence ever be Cauchy. 00:09:05.440 --> 00:09:08.590 align:middle line:84% And therefore, this has no convergent subsequence. 00:09:08.590 --> 00:09:12.640 align:middle line:84% So the identity on little l2 is not a compact operator. 00:09:12.640 --> 00:09:23.150 align:middle line:84% And by this argument, you can, in fact, 00:09:23.150 --> 00:09:25.730 align:middle line:84% prove that the identity is not a compact operator 00:09:25.730 --> 00:09:28.055 align:middle line:84% on any infinite-dimensional Hilbert space. 00:09:28.055 --> 00:09:35.450 align:middle line:90% 00:09:35.450 --> 00:09:41.610 align:middle line:84% So we have one way of showing that an operator is 00:09:41.610 --> 00:09:45.510 align:middle line:84% a compact operator, which is by verifying the definition. 00:09:45.510 --> 00:09:47.380 align:middle line:84% And then we now have this theorem, 00:09:47.380 --> 00:09:49.740 align:middle line:84% which I alluded to at the start, showing 00:09:49.740 --> 00:09:55.140 align:middle line:84% that the set of compact operators 00:09:55.140 --> 00:10:00.470 align:middle line:84% is the closure of the finite rank operators. 00:10:00.470 --> 00:10:14.130 align:middle line:84% So let H be a separable Hilbert space. 00:10:14.130 --> 00:10:18.850 align:middle line:90% 00:10:18.850 --> 00:10:27.850 align:middle line:84% Then, an operator T, bounded linear operator T on H, 00:10:27.850 --> 00:10:36.820 align:middle line:84% is compact operator if and only if there exists a sequence T 00:10:36.820 --> 00:10:49.210 align:middle line:84% n of finite rank ops, which converge to T 00:10:49.210 --> 00:10:50.830 align:middle line:90% n the operator norm. 00:10:50.830 --> 00:10:57.390 align:middle line:90% 00:10:57.390 --> 00:11:00.620 align:middle line:84% So this proves that the set of compact operators 00:11:00.620 --> 00:11:06.320 align:middle line:84% is the closure of the set of finite rank operators, which 00:11:06.320 --> 00:11:07.295 align:middle line:90% we denoted by-- 00:11:07.295 --> 00:11:09.940 align:middle line:90% 00:11:09.940 --> 00:11:14.030 align:middle line:84% so this shows compact operators is 00:11:14.030 --> 00:11:16.310 align:middle line:84% the closure of the finite rank operators, which 00:11:16.310 --> 00:11:30.270 align:middle line:90% we denoted by R. That's R of H. 00:11:30.270 --> 00:11:42.020 align:middle line:84% So first direction is to show that if I 00:11:42.020 --> 00:11:43.580 align:middle line:84% have a compact operator, then it's 00:11:43.580 --> 00:11:49.480 align:middle line:84% equal to a norm limit of finite rank operators. 00:11:49.480 --> 00:11:54.055 align:middle line:84% So since H is separable, it has an orthonormal basis. 00:11:54.055 --> 00:12:05.110 align:middle line:90% 00:12:05.110 --> 00:12:22.390 align:middle line:84% Now, since T is a compact operator, 00:12:22.390 --> 00:12:25.840 align:middle line:84% remember this means that the image 00:12:25.840 --> 00:12:32.590 align:middle line:84% T u with of all those elements of norm less than or equal to 1 00:12:32.590 --> 00:12:48.785 align:middle line:84% is contained in a compact set, namely the closure of this set. 00:12:48.785 --> 00:12:51.380 align:middle line:90% 00:12:51.380 --> 00:12:57.740 align:middle line:84% Now, since this set is contained in a compact set, I can use-- 00:12:57.740 --> 00:13:12.280 align:middle line:84% in fact, let me just put it's a compact set. 00:13:12.280 --> 00:13:13.780 align:middle line:84% And so we have this characterization 00:13:13.780 --> 00:13:18.070 align:middle line:84% of compact subsets of a separable Hilbert space 00:13:18.070 --> 00:13:23.170 align:middle line:84% by being closed, bounded, and having equismall tails. 00:13:23.170 --> 00:13:29.660 align:middle line:84% So then, for every epsilon positive, 00:13:29.660 --> 00:13:33.550 align:middle line:84% there exists a natural number N such 00:13:33.550 --> 00:13:51.880 align:middle line:84% that if I look at sum over K bigger than N, K u e K squared, 00:13:51.880 --> 00:13:58.000 align:middle line:84% this is less than epsilon squared for all elements 00:13:58.000 --> 00:13:59.680 align:middle line:84% with norm less than or equal to 1. 00:13:59.680 --> 00:14:02.740 align:middle line:84% Because the K u's-- these are in this compact-- 00:14:02.740 --> 00:14:03.990 align:middle line:90% why am I using K? 00:14:03.990 --> 00:14:06.340 align:middle line:90% Should use T, sorry. 00:14:06.340 --> 00:14:08.620 align:middle line:84% The T u's are all in this compact set. 00:14:08.620 --> 00:14:10.840 align:middle line:84% So we have this equismall tail condition 00:14:10.840 --> 00:14:13.210 align:middle line:84% that for every epsilon, we can find 00:14:13.210 --> 00:14:17.170 align:middle line:84% a N which does not depend on the elements of the compact set, 00:14:17.170 --> 00:14:21.175 align:middle line:84% so that the tails of the Fourier coefficients are small. 00:14:21.175 --> 00:14:26.450 align:middle line:90% 00:14:26.450 --> 00:14:28.820 align:middle line:84% So what this says is that we can basically, 00:14:28.820 --> 00:14:32.030 align:middle line:84% if we're taking any u with norm less than or equal to 1 00:14:32.030 --> 00:14:40.190 align:middle line:84% and hitting it with T, then the end of the Fourier coefficients 00:14:40.190 --> 00:14:41.610 align:middle line:90% don't really matter. 00:14:41.610 --> 00:14:43.580 align:middle line:84% And so we should think we can approximate T 00:14:43.580 --> 00:14:47.630 align:middle line:84% by just a finite rank operator that consists of coefficients 00:14:47.630 --> 00:14:50.750 align:middle line:84% like this times e k for k between 1 to N. 00:14:50.750 --> 00:14:53.790 align:middle line:84% And that's basically what we're going to do. 00:14:53.790 --> 00:15:07.630 align:middle line:84% So for n, define T n of u to be sum from k equals 1 to n 00:15:07.630 --> 00:15:12.430 align:middle line:90% of T u e k e k. 00:15:12.430 --> 00:15:19.410 align:middle line:90% 00:15:19.410 --> 00:15:24.660 align:middle line:84% And this is for u in H. So one can 00:15:24.660 --> 00:15:30.030 align:middle line:84% check that this is a bounded linear operator 00:15:30.030 --> 00:15:31.320 align:middle line:90% and it's also a finite rank. 00:15:31.320 --> 00:15:33.015 align:middle line:90% Because the range is-- 00:15:33.015 --> 00:15:37.630 align:middle line:90% 00:15:37.630 --> 00:15:40.210 align:middle line:84% so it's clearly a bounded linear operator. 00:15:40.210 --> 00:15:44.065 align:middle line:84% Just take the norm squared of this. 00:15:44.065 --> 00:15:46.312 align:middle line:84% This is equal to the sum of squares of the Fourier 00:15:46.312 --> 00:15:48.520 align:middle line:84% coefficients, which is less than or equal to the norm 00:15:48.520 --> 00:15:51.370 align:middle line:84% of 2 squared, which is less than or equal to the norm 00:15:51.370 --> 00:15:52.370 align:middle line:90% of T squared. 00:15:52.370 --> 00:15:56.270 align:middle line:84% So it's clearly a bounded linear operator. 00:15:56.270 --> 00:16:00.670 align:middle line:84% And if I look at the range of T n, 00:16:00.670 --> 00:16:06.190 align:middle line:84% this is a subset of the span of e1 up to e n. 00:16:06.190 --> 00:16:13.490 align:middle line:84% And therefore, T n is a finite rank operator for all n. 00:16:13.490 --> 00:16:16.750 align:middle line:90% 00:16:16.750 --> 00:16:21.970 align:middle line:84% So I claim this is our set of finite rank operators 00:16:21.970 --> 00:16:30.310 align:middle line:84% that converges to T as n goes to infinity. 00:16:30.310 --> 00:16:35.940 align:middle line:84% So to prove this, we'll just do standard epsilon N argument. 00:16:35.940 --> 00:16:38.790 align:middle line:90% Let epsilon be positive. 00:16:38.790 --> 00:16:50.200 align:middle line:84% Then let N be the N that appears in this condition star up here, 00:16:50.200 --> 00:16:56.980 align:middle line:84% meaning capital N is the natural numbers, guaranteeing this. 00:16:56.980 --> 00:17:01.470 align:middle line:90% 00:17:01.470 --> 00:17:04.040 align:middle line:84% So now, I claim that for all little n bigger than 00:17:04.040 --> 00:17:08.060 align:middle line:84% or equal to this capital N, the operator norm of T n minus T 00:17:08.060 --> 00:17:10.610 align:middle line:84% is less than or equal to epsilon, 00:17:10.610 --> 00:17:11.735 align:middle line:90% which is still good enough. 00:17:11.735 --> 00:17:20.660 align:middle line:90% 00:17:20.660 --> 00:17:25.790 align:middle line:84% Then for all-- then if norm of u equals 1, 00:17:25.790 --> 00:17:31.482 align:middle line:84% we compute that T n of u minus T u. 00:17:31.482 --> 00:17:35.690 align:middle line:90% 00:17:35.690 --> 00:17:39.290 align:middle line:84% So we haven't used anything about this e k yet, 00:17:39.290 --> 00:17:42.380 align:middle line:84% but this is where we use that they're in orthonormal basis, 00:17:42.380 --> 00:17:44.960 align:middle line:90% is that I can write T n of u. 00:17:44.960 --> 00:17:47.360 align:middle line:84% Well, first off, this just is from the definition 00:17:47.360 --> 00:17:49.430 align:middle line:90% that this is K equals 1 to n-- 00:17:49.430 --> 00:17:53.120 align:middle line:90% and let me put a square here-- 00:17:53.120 --> 00:17:59.130 align:middle line:90% of T u E k E k minus-- 00:17:59.130 --> 00:18:06.590 align:middle line:84% and now I expand this element as in this basis T u 00:18:06.590 --> 00:18:10.685 align:middle line:90% e k e k squared. 00:18:10.685 --> 00:18:13.760 align:middle line:90% 00:18:13.760 --> 00:18:17.190 align:middle line:84% You see what cancels out are the first n of these. 00:18:17.190 --> 00:18:20.430 align:middle line:90% 00:18:20.430 --> 00:18:30.036 align:middle line:84% And this equals the norm squared of k bigger than n of T u e k 00:18:30.036 --> 00:18:33.070 align:middle line:90% e k norm squared. 00:18:33.070 --> 00:18:38.340 align:middle line:84% And this is equal to, since these are orthogonal, 00:18:38.340 --> 00:18:39.840 align:middle line:90% k bigger than e-- 00:18:39.840 --> 00:18:48.430 align:middle line:84% orthonormal, i mean-- k bigger than n T u j k squared. 00:18:48.430 --> 00:18:52.410 align:middle line:84% And now, n is bigger than or equal to capital N. 00:18:52.410 --> 00:18:56.070 align:middle line:84% So this is less than or equal to k 00:18:56.070 --> 00:19:02.820 align:middle line:84% bigger than N T u e k squared, and that's 00:19:02.820 --> 00:19:04.000 align:middle line:90% less than epsilon squared. 00:19:04.000 --> 00:19:09.870 align:middle line:84% So what we've shown is that for all u of norm 1, 00:19:09.870 --> 00:19:14.430 align:middle line:84% T N u minus T u squared is less than epsilon squared. 00:19:14.430 --> 00:19:19.800 align:middle line:84% And therefore, if I take the sup over all u that have norm 1, 00:19:19.800 --> 00:19:23.580 align:middle line:84% I get that the norm of T n minus T, which is that sup, 00:19:23.580 --> 00:19:27.280 align:middle line:84% is less than or equal to epsilon. 00:19:27.280 --> 00:19:37.300 align:middle line:84% So thus, the T n's converge to T in operator norm. 00:19:37.300 --> 00:19:47.170 align:middle line:90% 00:19:47.170 --> 00:19:49.630 align:middle line:84% So now, going in the opposite direction, 00:19:49.630 --> 00:19:55.590 align:middle line:84% we will use this second characterization 00:19:55.590 --> 00:20:00.120 align:middle line:84% that we have of compact sets that we didn't prove, 00:20:00.120 --> 00:20:02.460 align:middle line:84% but I said you should look up if you want to. 00:20:02.460 --> 00:20:04.360 align:middle line:84% It's, again, a diagonalization argument, 00:20:04.360 --> 00:20:07.710 align:middle line:84% but that says that compact subsets can be approximated 00:20:07.710 --> 00:20:10.200 align:middle line:90% by finite dimensional-- 00:20:10.200 --> 00:20:14.480 align:middle line:90% or are sets that are compact-- 00:20:14.480 --> 00:20:16.460 align:middle line:84% or a set is compact if and only if it 00:20:16.460 --> 00:20:22.070 align:middle line:84% can be approximated by finite-dimensional subspaces. 00:20:22.070 --> 00:20:27.590 align:middle line:84% So now, we're proving the opposite direction. 00:20:27.590 --> 00:20:37.780 align:middle line:84% And suppose that T can be approximated in operator norm 00:20:37.780 --> 00:20:40.270 align:middle line:84% by a sequence of finite rank operators. 00:20:40.270 --> 00:20:47.650 align:middle line:90% 00:20:47.650 --> 00:20:50.020 align:middle line:84% And we want to show that T is a compact operator. 00:20:50.020 --> 00:20:58.390 align:middle line:90% 00:20:58.390 --> 00:21:07.800 align:middle line:84% So first, note that if I look at T u norm less than 00:21:07.800 --> 00:21:09.880 align:middle line:90% or equal to 1 closure-- 00:21:09.880 --> 00:21:12.450 align:middle line:84% so this is the set that I'm trying 00:21:12.450 --> 00:21:16.380 align:middle line:84% to show is a compact set-- it's obviously closed because it's 00:21:16.380 --> 00:21:18.090 align:middle line:90% the closure of a set. 00:21:18.090 --> 00:21:21.240 align:middle line:84% The closure of a set is always the smallest closed set 00:21:21.240 --> 00:21:23.210 align:middle line:90% containing that set. 00:21:23.210 --> 00:21:30.320 align:middle line:84% And it's contained because T is a bounded operator. 00:21:30.320 --> 00:21:33.860 align:middle line:84% It's always contained in the set of, 00:21:33.860 --> 00:21:38.090 align:middle line:84% let's call it v in H, where the norm of v 00:21:38.090 --> 00:21:43.460 align:middle line:84% is less than or equal to the norm of T. 00:21:43.460 --> 00:22:01.690 align:middle line:84% So this shows that the set here is closed and bounded. 00:22:01.690 --> 00:22:09.290 align:middle line:90% 00:22:09.290 --> 00:22:11.870 align:middle line:84% So to conclude that this set is compact, we just 00:22:11.870 --> 00:22:13.820 align:middle line:84% need to show that it can be approximated 00:22:13.820 --> 00:22:17.640 align:middle line:84% by finite-dimensional subspaces, meaning for every epsilon, 00:22:17.640 --> 00:22:19.680 align:middle line:84% there exists a finite-dimensional subspace 00:22:19.680 --> 00:22:23.060 align:middle line:84% so that the distance from every element 00:22:23.060 --> 00:22:28.655 align:middle line:84% in this set to the subspace W is less than epsilon. 00:22:28.655 --> 00:22:33.040 align:middle line:90% 00:22:33.040 --> 00:22:43.090 align:middle line:84% So we want to show now, or we make this a claim, 00:22:43.090 --> 00:22:48.130 align:middle line:84% for all epsilon positive, there exists 00:22:48.130 --> 00:23:02.820 align:middle line:84% a finite-dimensional subspace W such that for all u less than 00:23:02.820 --> 00:23:11.480 align:middle line:84% or equal to 1, we have that the distance from this element T 00:23:11.480 --> 00:23:19.100 align:middle line:90% u to w is less than epsilon. 00:23:19.100 --> 00:23:23.330 align:middle line:84% That's what I mean by we can approximate this set 00:23:23.330 --> 00:23:27.200 align:middle line:90% by finite-dimensional subspaces. 00:23:27.200 --> 00:23:31.370 align:middle line:84% Once we can do that, then we've proven the claim, 00:23:31.370 --> 00:23:33.890 align:middle line:84% or we can use that characterization 00:23:33.890 --> 00:23:37.850 align:middle line:84% of compact sets to conclude that this set is compact. 00:23:37.850 --> 00:23:39.860 align:middle line:90% Now, what's the idea? 00:23:39.860 --> 00:23:43.850 align:middle line:84% We have that T, which again, this set here is just 00:23:43.850 --> 00:23:48.780 align:middle line:84% the image of the closed unit ball by T taking the closure. 00:23:48.780 --> 00:23:52.280 align:middle line:84% And therefore, T can be approximated by a finite rank 00:23:52.280 --> 00:23:53.000 align:middle line:90% operator. 00:23:53.000 --> 00:23:54.990 align:middle line:90% Its range is finite dimensional. 00:23:54.990 --> 00:23:58.880 align:middle line:84% So this set should be approximated 00:23:58.880 --> 00:24:03.840 align:middle line:84% by the range of T n, which is finite dimensional, 00:24:03.840 --> 00:24:06.390 align:middle line:84% and therefore, should give us our claim. 00:24:06.390 --> 00:24:08.290 align:middle line:90% And that's exactly what we do. 00:24:08.290 --> 00:24:16.390 align:middle line:84% So since the norm of T n minus T goes to 0, 00:24:16.390 --> 00:24:20.110 align:middle line:84% there exists a natural number N such 00:24:20.110 --> 00:24:25.625 align:middle line:84% that I have that for all little n bigger 00:24:25.625 --> 00:24:27.500 align:middle line:84% than or equal to capital N, blah, blah, blah. 00:24:27.500 --> 00:24:29.260 align:middle line:90% But I don't need all little n. 00:24:29.260 --> 00:24:32.485 align:middle line:84% I just have this less than epsilon. 00:24:32.485 --> 00:24:40.007 align:middle line:90% 00:24:40.007 --> 00:24:47.150 align:middle line:84% Let W be the range of T sub n, which is 00:24:47.150 --> 00:24:48.455 align:middle line:90% a finite-dimensional subspace. 00:24:48.455 --> 00:24:57.400 align:middle line:90% 00:24:57.400 --> 00:25:00.720 align:middle line:84% And now, I want to show that I have this condition here-- 00:25:00.720 --> 00:25:03.210 align:middle line:84% that for all u with norm less than or equal to 1, 00:25:03.210 --> 00:25:08.152 align:middle line:84% this distance from T u to capital W is less than-- 00:25:08.152 --> 00:25:10.110 align:middle line:84% we can make that less than or equal to epsilon. 00:25:10.110 --> 00:25:13.170 align:middle line:84% That still proves what we want, just 00:25:13.170 --> 00:25:15.450 align:middle line:84% go through and change epsilon to epsilon over 2. 00:25:15.450 --> 00:25:19.010 align:middle line:90% 00:25:19.010 --> 00:25:26.240 align:middle line:84% Then, for all elements in H with norm less than or equal to 1, 00:25:26.240 --> 00:25:32.690 align:middle line:84% if I take T u and subtract off T n u, which 00:25:32.690 --> 00:25:39.820 align:middle line:84% is an element in capital W, this is 00:25:39.820 --> 00:25:43.630 align:middle line:90% less than or equal to T minus-- 00:25:43.630 --> 00:25:46.600 align:middle line:84% so the operator norm of T minus T 00:25:46.600 --> 00:25:53.590 align:middle line:84% n times norm of u, which is less than or equal to T minus T 00:25:53.590 --> 00:25:57.180 align:middle line:90% n, which is less than epsilon. 00:25:57.180 --> 00:26:03.010 align:middle line:90% 00:26:03.010 --> 00:26:04.760 align:middle line:84% Put a less than there-- it doesn't matter. 00:26:04.760 --> 00:26:07.470 align:middle line:90% 00:26:07.470 --> 00:26:11.030 align:middle line:84% So this holds for one particular element of W, 00:26:11.030 --> 00:26:17.210 align:middle line:84% and therefore, the n-th, which is the distance from T u 00:26:17.210 --> 00:26:20.642 align:middle line:84% to capital W, which is less than or equal to this, 00:26:20.642 --> 00:26:21.725 align:middle line:90% must be less than epsilon. 00:26:21.725 --> 00:26:25.010 align:middle line:90% 00:26:25.010 --> 00:26:30.600 align:middle line:84% And therefore, this proves the claim. 00:26:30.600 --> 00:26:34.973 align:middle line:84% And the fact that this set is compact 00:26:34.973 --> 00:26:36.390 align:middle line:84% follows from that characterization 00:26:36.390 --> 00:26:39.250 align:middle line:84% of compact sets, which we proved-- 00:26:39.250 --> 00:26:42.520 align:middle line:84% or didn't prove, but at least stated a few lectures ago. 00:26:42.520 --> 00:26:46.810 align:middle line:90% 00:26:46.810 --> 00:26:49.680 align:middle line:84% So I mean, you can use the definition to verify something 00:26:49.680 --> 00:26:51.390 align:middle line:90% as a compact operator. 00:26:51.390 --> 00:26:54.840 align:middle line:84% You could use this alternative characterization 00:26:54.840 --> 00:27:00.210 align:middle line:84% as a limit in operator norm of finite rank operators 00:27:00.210 --> 00:27:03.960 align:middle line:84% to verify something, say compact operator. 00:27:03.960 --> 00:27:06.540 align:middle line:90% 00:27:06.540 --> 00:27:09.630 align:middle line:84% It's just whatever's most convenient at the time. 00:27:09.630 --> 00:27:12.420 align:middle line:90% So how about the end of-- 00:27:12.420 --> 00:27:14.340 align:middle line:84% when we talked about finite rank operators, 00:27:14.340 --> 00:27:19.200 align:middle line:84% we also went a little bit into their algebraic structure. 00:27:19.200 --> 00:27:21.030 align:middle line:84% We can do the same for compact operators. 00:27:21.030 --> 00:27:31.790 align:middle line:84% So again, let H be a separable Hilbert space. 00:27:31.790 --> 00:27:37.520 align:middle line:84% Then the first property of compact operators is that-- 00:27:37.520 --> 00:27:50.150 align:middle line:84% and let's call it a name, K of H be the set of compact operators 00:27:50.150 --> 00:27:53.030 align:middle line:90% on H. 00:27:53.030 --> 00:28:02.980 align:middle line:84% And the first is K of H is a closed 00:28:02.980 --> 00:28:10.990 align:middle line:84% subspace of the space of bounded linear operators on the Hilbert 00:28:10.990 --> 00:28:12.380 align:middle line:90% space. 00:28:12.380 --> 00:28:14.680 align:middle line:84% So the fact that it's a linear subspace 00:28:14.680 --> 00:28:17.215 align:middle line:90% is not too difficult to prove. 00:28:17.215 --> 00:28:19.090 align:middle line:84% The fact that it's closed follows immediately 00:28:19.090 --> 00:28:20.840 align:middle line:84% from what we've done so far when we showed 00:28:20.840 --> 00:28:23.980 align:middle line:84% that the compact operators are the closure of the set 00:28:23.980 --> 00:28:26.943 align:middle line:90% of finite rank operators. 00:28:26.943 --> 00:28:28.360 align:middle line:84% Really, I mean the fact that we've 00:28:28.360 --> 00:28:30.310 align:middle line:84% shown that proves both of these, because the closure 00:28:30.310 --> 00:28:31.925 align:middle line:84% of a subspace is, again, a subspace. 00:28:31.925 --> 00:28:39.120 align:middle line:90% 00:28:39.120 --> 00:28:44.430 align:middle line:84% If T is a compact operator, then its adjoint 00:28:44.430 --> 00:28:46.230 align:middle line:90% is also a compact operator. 00:28:46.230 --> 00:28:49.380 align:middle line:90% 00:28:49.380 --> 00:28:58.220 align:middle line:84% And for every pair of bounded linear operators 00:28:58.220 --> 00:29:03.500 align:middle line:84% and compact operator, if I multiply on the left 00:29:03.500 --> 00:29:09.050 align:middle line:84% and on the right by these bounded linear operators, 00:29:09.050 --> 00:29:14.160 align:middle line:90% that remains a compact operator. 00:29:14.160 --> 00:29:16.850 align:middle line:84% So a fancy way of combining 2 and 3 00:29:16.850 --> 00:29:22.050 align:middle line:84% is that the set of compact operators 00:29:22.050 --> 00:29:27.330 align:middle line:84% is a star closed two-sided ideal in the algebra bounded 00:29:27.330 --> 00:29:29.960 align:middle line:90% linear operators. 00:29:29.960 --> 00:29:37.100 align:middle line:84% But I'm just going to list conditions, and not 00:29:37.100 --> 00:29:38.610 align:middle line:90% give fancy words for them. 00:29:38.610 --> 00:29:39.580 align:middle line:90% So 1 is clear. 00:29:39.580 --> 00:29:41.330 align:middle line:84% I'm not going to write the proof for that. 00:29:41.330 --> 00:29:43.430 align:middle line:90% We'll just do 2 and 3. 00:29:43.430 --> 00:29:47.450 align:middle line:84% And although we could do it directly from the definition, 00:29:47.450 --> 00:29:51.330 align:middle line:84% let's use the theorem that we just worked hard to prove. 00:29:51.330 --> 00:29:57.380 align:middle line:84% So if T is a compact operator, then 00:29:57.380 --> 00:30:01.850 align:middle line:84% that implies that there exists a sequence of finite rank 00:30:01.850 --> 00:30:08.450 align:middle line:84% operators such that T n converges 00:30:08.450 --> 00:30:11.210 align:middle line:90% to T in operator norm. 00:30:11.210 --> 00:30:17.620 align:middle line:84% And therefore, by what we proved last time, 00:30:17.620 --> 00:30:19.900 align:middle line:84% we showed that if we have a finite rank operator, 00:30:19.900 --> 00:30:26.400 align:middle line:84% then its adjoint is also a finite rank operator, which 00:30:26.400 --> 00:30:36.920 align:middle line:84% implies that if I take the norm of the adjoint 00:30:36.920 --> 00:30:41.800 align:middle line:84% of this finite rank operator and minus 00:30:41.800 --> 00:30:45.310 align:middle line:84% the adjoint of the compact operator-- remember, 00:30:45.310 --> 00:30:47.950 align:middle line:84% we have this nice theorem that tells us 00:30:47.950 --> 00:30:50.590 align:middle line:84% the norm of the adjoint is equal to the norm 00:30:50.590 --> 00:30:58.180 align:middle line:84% of the original operator-- this is equal to T n minus T, which 00:30:58.180 --> 00:31:00.400 align:middle line:90% we're assuming goes to 0. 00:31:00.400 --> 00:31:04.960 align:middle line:84% And therefore, the adjoint is a norm limit 00:31:04.960 --> 00:31:08.420 align:middle line:90% of finite rank operators. 00:31:08.420 --> 00:31:10.700 align:middle line:84% And therefore, it must be a compact operator. 00:31:10.700 --> 00:31:15.660 align:middle line:90% 00:31:15.660 --> 00:31:19.800 align:middle line:84% And to prove 3, we'll use this characterization again. 00:31:19.800 --> 00:31:27.900 align:middle line:84% So again, assume T is a compact operator so that there exists 00:31:27.900 --> 00:31:31.680 align:middle line:84% a sequence of T n's and a finite rank operators 00:31:31.680 --> 00:31:34.470 align:middle line:84% converging to T and the operator norm. 00:31:34.470 --> 00:31:38.760 align:middle line:90% 00:31:38.760 --> 00:31:47.310 align:middle line:84% So T n finite rank operators, T n minus T 00:31:47.310 --> 00:31:55.350 align:middle line:90% converges to 0 implies, then-- 00:31:55.350 --> 00:31:58.860 align:middle line:84% now, last time, we showed that the set of finite rank 00:31:58.860 --> 00:32:01.600 align:middle line:84% operators also satisfies these two conditions. 00:32:01.600 --> 00:32:04.140 align:middle line:84% This one I just used here, but also satisfies 00:32:04.140 --> 00:32:06.750 align:middle line:84% this one-- that if I take a finite rank operator 00:32:06.750 --> 00:32:09.600 align:middle line:84% and multiply it on either side by a bounded linear operator, 00:32:09.600 --> 00:32:14.770 align:middle line:84% I still have a finite rank operator. 00:32:14.770 --> 00:32:18.420 align:middle line:84% So for all n, I also have A times 00:32:18.420 --> 00:32:22.450 align:middle line:84% T n B is also a finite rank operator 00:32:22.450 --> 00:32:24.210 align:middle line:90% by what we did last time. 00:32:24.210 --> 00:32:31.860 align:middle line:84% And if I take the norm of A T n B minus A T B, 00:32:31.860 --> 00:32:42.330 align:middle line:84% this is equal to the norm of A T n minus T B. And now, 00:32:42.330 --> 00:32:46.410 align:middle line:84% the norm preserves this kind of algebraic structure 00:32:46.410 --> 00:32:48.240 align:middle line:90% of bounded linear operators-- 00:32:48.240 --> 00:32:50.850 align:middle line:84% I shouldn't say "preserves," but "respects" it, 00:32:50.850 --> 00:32:54.300 align:middle line:84% meaning the norm of a product is less than 00:32:54.300 --> 00:32:56.080 align:middle line:84% or equal to the product of the norm. 00:32:56.080 --> 00:32:57.720 align:middle line:84% So this is less than or equal to-- 00:32:57.720 --> 00:33:01.420 align:middle line:84% this just follows from the definition-- 00:33:01.420 --> 00:33:08.460 align:middle line:84% T n minus T B. And so this norm is fixed. 00:33:08.460 --> 00:33:09.690 align:middle line:90% This norm is fixed. 00:33:09.690 --> 00:33:13.330 align:middle line:90% This thing is going to 0. 00:33:13.330 --> 00:33:16.930 align:middle line:84% So I have bounded above this non-negative quantity 00:33:16.930 --> 00:33:18.850 align:middle line:90% by something converging to 0. 00:33:18.850 --> 00:33:22.510 align:middle line:84% And therefore, this thing converges to 0. 00:33:22.510 --> 00:33:25.600 align:middle line:84% So A times T times B is the norm limit 00:33:25.600 --> 00:33:27.490 align:middle line:84% of a sequence of finite rank operators. 00:33:27.490 --> 00:33:28.930 align:middle line:90% So it's also compact. 00:33:28.930 --> 00:33:45.670 align:middle line:90% 00:33:45.670 --> 00:33:50.140 align:middle line:90% Now, coming back to-- 00:33:50.140 --> 00:33:56.600 align:middle line:90% 00:33:56.600 --> 00:33:58.970 align:middle line:90% one second. 00:33:58.970 --> 00:33:59.855 align:middle line:90% Hydration is key. 00:33:59.855 --> 00:34:20.870 align:middle line:90% 00:34:20.870 --> 00:34:25.610 align:middle line:84% Now, one of the most important set 00:34:25.610 --> 00:34:32.900 align:middle line:84% of numbers that one can associate to a matrix 00:34:32.900 --> 00:34:43.790 align:middle line:84% is the set of its eigenvalues, and not just-- 00:34:43.790 --> 00:34:50.630 align:middle line:84% so these eigenvalues typically represent possibly the modes 00:34:50.630 --> 00:34:53.900 align:middle line:90% of vibration of a string. 00:34:53.900 --> 00:34:57.710 align:middle line:84% Or if you're taking Quantum Mechanics, in which case 00:34:57.710 --> 00:35:01.010 align:middle line:84% you're not necessarily dealing with matrices, depending 00:35:01.010 --> 00:35:04.370 align:middle line:84% on how it's being taught to you, these eigenvalues 00:35:04.370 --> 00:35:06.785 align:middle line:84% are the energy levels for bound states. 00:35:06.785 --> 00:35:10.240 align:middle line:90% 00:35:10.240 --> 00:35:14.530 align:middle line:84% So perhaps, since compact operators 00:35:14.530 --> 00:35:18.520 align:middle line:84% are the inverses of differential operators, which 00:35:18.520 --> 00:35:20.410 align:middle line:84% most definitely appear in quantum mechanics 00:35:20.410 --> 00:35:21.850 align:middle line:84% but also in other applications, it 00:35:21.850 --> 00:35:28.120 align:middle line:84% would be good to develop some sort of theory of eigenvalues 00:35:28.120 --> 00:35:30.820 align:middle line:84% and eigenvectors for more general 00:35:30.820 --> 00:35:33.700 align:middle line:84% bounded linear operators, and then 00:35:33.700 --> 00:35:35.260 align:middle line:90% specify to compact operators. 00:35:35.260 --> 00:35:38.210 align:middle line:90% But what am I going on about? 00:35:38.210 --> 00:35:40.690 align:middle line:84% So now, we're going to discuss the spectrum 00:35:40.690 --> 00:35:50.480 align:middle line:84% of a bounded linear operator, which 00:35:50.480 --> 00:36:00.010 align:middle line:84% is a generalization of eigenvalues and eigenvectors 00:36:00.010 --> 00:36:03.100 align:middle line:84% that you encountered in linear algebra. 00:36:03.100 --> 00:36:08.840 align:middle line:90% 00:36:08.840 --> 00:36:12.990 align:middle line:84% In linear algebra, the spectrum will, 00:36:12.990 --> 00:36:17.760 align:middle line:84% if I take H to be just a finite-dimensional space of R n 00:36:17.760 --> 00:36:21.570 align:middle line:84% C n, then the spectrum, as we're going to introduce it now, 00:36:21.570 --> 00:36:26.220 align:middle line:84% consists entirely of the eigenvalues of that matrix. 00:36:26.220 --> 00:36:31.990 align:middle line:84% Not so if we move on to now bounded linear operators, which 00:36:31.990 --> 00:36:33.850 align:middle line:84% is what makes it pretty interesting, 00:36:33.850 --> 00:36:39.490 align:middle line:84% is that there's more stuff that's associated to the fact 00:36:39.490 --> 00:36:41.410 align:middle line:84% that you're working in infinite dimensions. 00:36:41.410 --> 00:36:44.620 align:middle line:90% 00:36:44.620 --> 00:36:47.770 align:middle line:84% And rather than harp on about it, let's go. 00:36:47.770 --> 00:36:52.760 align:middle line:84% Let's just dive in and start discussing the spectrum. 00:36:52.760 --> 00:36:56.630 align:middle line:84% First, I want to bring up a few facts that you proved in, 00:36:56.630 --> 00:36:58.130 align:middle line:84% I think it was the first assignment, 00:36:58.130 --> 00:37:01.800 align:middle line:84% first or second assignment of the course. 00:37:01.800 --> 00:37:05.660 align:middle line:84% So throughout, again, H is a Hilbert space. 00:37:05.660 --> 00:37:08.510 align:middle line:84% I will say it maybe at one point that it's separable, 00:37:08.510 --> 00:37:12.390 align:middle line:84% but throughout, H is always a Hilbert space. 00:37:12.390 --> 00:37:17.030 align:middle line:84% So we have the following theorem, which tells you 00:37:17.030 --> 00:37:19.070 align:middle line:84% how to invert certain operators via what's 00:37:19.070 --> 00:37:20.000 align:middle line:90% called Neumann series. 00:37:20.000 --> 00:37:22.835 align:middle line:90% 00:37:22.835 --> 00:37:29.570 align:middle line:84% Let T be a bounded linear operator on a Hilbert space. 00:37:29.570 --> 00:37:40.430 align:middle line:84% If the norm of T is less than 1, then the operator I minus T 00:37:40.430 --> 00:37:41.750 align:middle line:90% is invertible. 00:37:41.750 --> 00:37:44.750 align:middle line:90% 00:37:44.750 --> 00:37:47.510 align:middle line:84% And if I want to compute the inverse, 00:37:47.510 --> 00:37:51.110 align:middle line:84% I can compute it just like I would 00:37:51.110 --> 00:37:55.430 align:middle line:90% if I minus T was 1 minus x. 00:37:55.430 --> 00:38:00.820 align:middle line:84% And we're talking about dividing by the function 1 minus x. 00:38:00.820 --> 00:38:04.187 align:middle line:90% You have a power series-- 00:38:04.187 --> 00:38:05.770 align:middle line:84% not power series, but you can write it 00:38:05.770 --> 00:38:07.540 align:middle line:90% as a geometric series-- 00:38:07.540 --> 00:38:11.590 align:middle line:84% sum from n equals 0 to infinity T to the n. 00:38:11.590 --> 00:38:15.170 align:middle line:84% And this series over here is absolutely convergent. 00:38:15.170 --> 00:38:15.670 align:middle line:90% Why? 00:38:15.670 --> 00:38:18.393 align:middle line:90% Because the norm of every term-- 00:38:18.393 --> 00:38:20.560 align:middle line:84% norm of T to the n is less than or equal to the norm 00:38:20.560 --> 00:38:22.570 align:middle line:90% of T raised to the n. 00:38:22.570 --> 00:38:24.250 align:middle line:84% And the norm of t is less than 1. 00:38:24.250 --> 00:38:28.780 align:middle line:84% So this thing is absolutely summable. 00:38:28.780 --> 00:38:32.020 align:middle line:84% And you use that when you actually proved this theorem 00:38:32.020 --> 00:38:32.960 align:middle line:90% in the assignment. 00:38:32.960 --> 00:38:36.630 align:middle line:84% So I'm not going to give the proof of it again. 00:38:36.630 --> 00:38:38.310 align:middle line:84% I didn't give it once before, you did, 00:38:38.310 --> 00:38:40.590 align:middle line:90% but I'm not going to prove it. 00:38:40.590 --> 00:38:43.290 align:middle line:90% We'll just move on. 00:38:43.290 --> 00:38:45.000 align:middle line:84% And so another fact that you proved 00:38:45.000 --> 00:38:52.230 align:middle line:84% using this is that the set GL h, which 00:38:52.230 --> 00:39:01.430 align:middle line:84% is the set of bounded linear operators that are invertible, 00:39:01.430 --> 00:39:03.380 align:middle line:90% meaning it's bijective-- 00:39:03.380 --> 00:39:05.180 align:middle line:84% by the open mapping theorem, we know 00:39:05.180 --> 00:39:08.000 align:middle line:84% that if an operator is bijective, then 00:39:08.000 --> 00:39:14.840 align:middle line:84% it automatically has a bounded inverse, 00:39:14.840 --> 00:39:17.780 align:middle line:90% so I'll say bijective-- 00:39:17.780 --> 00:39:25.640 align:middle line:84% that this is an open set, is an open subset of what? 00:39:25.640 --> 00:39:29.660 align:middle line:84% Of the space of bounded linear operators. 00:39:29.660 --> 00:39:33.190 align:middle line:90% 00:39:33.190 --> 00:39:40.580 align:middle line:84% So the proof is quite quick using this previous theorem. 00:39:40.580 --> 00:39:48.850 align:middle line:84% So let T0 be an element of the invertible operators on H, 00:39:48.850 --> 00:39:57.960 align:middle line:84% GL H. So that means we need to find an epsilon so 00:39:57.960 --> 00:40:01.770 align:middle line:84% that the ball centered at T0 of radius epsilon 00:40:01.770 --> 00:40:04.890 align:middle line:84% is contained in GL H, meaning everything in that ball 00:40:04.890 --> 00:40:06.030 align:middle line:90% is invertible. 00:40:06.030 --> 00:40:10.800 align:middle line:84% So suppose that I have an operator T 00:40:10.800 --> 00:40:20.990 align:middle line:84% and its distance in operator norm to T0 is less than 1 00:40:20.990 --> 00:40:23.855 align:middle line:84% over the operator norm of the inverse of T0. 00:40:23.855 --> 00:40:28.930 align:middle line:90% 00:40:28.930 --> 00:40:32.320 align:middle line:84% I claim that T then has to be invertible. 00:40:32.320 --> 00:40:34.930 align:middle line:90% 00:40:34.930 --> 00:40:44.380 align:middle line:84% Then, if I look at T0 times T minus T0 inverse-- 00:40:44.380 --> 00:40:49.060 align:middle line:84% so t0 inverse times T minus T0, in operator norm 00:40:49.060 --> 00:40:52.630 align:middle line:84% that's less than or equal to T norm of T inverse 00:40:52.630 --> 00:40:59.140 align:middle line:84% times the norm of T minus T0 and norm, and that's less than 1. 00:40:59.140 --> 00:41:07.300 align:middle line:84% So by the previous theorem I get that I minus T inverse T0 or T0 00:41:07.300 --> 00:41:14.160 align:middle line:84% inverse applied to T minus T0 is invertible 00:41:14.160 --> 00:41:18.670 align:middle line:84% because this thing here has norm less than 1, 00:41:18.670 --> 00:41:22.210 align:middle line:84% which implies that T, which is equal to the product of two 00:41:22.210 --> 00:41:30.400 align:middle line:84% invertible operators, namely T0 and I minus T0 minus 1, 00:41:30.400 --> 00:41:40.880 align:middle line:84% T minus T0, is a bijective bounded linear operator 00:41:40.880 --> 00:41:43.310 align:middle line:84% or a bounded linear operator with a bounded inverse. 00:41:43.310 --> 00:41:46.570 align:middle line:90% 00:41:46.570 --> 00:41:50.400 align:middle line:90% And so let me just summarize. 00:41:50.400 --> 00:42:04.500 align:middle line:84% I.e., the ball of radius 1 over the norm of the inverse 00:42:04.500 --> 00:42:10.170 align:middle line:84% is contained in the set of inverted 00:42:10.170 --> 00:42:12.540 align:middle line:90% bounded linear operators. 00:42:12.540 --> 00:42:14.910 align:middle line:84% And that proves that this is an open set. 00:42:14.910 --> 00:42:28.560 align:middle line:90% 00:42:28.560 --> 00:42:35.350 align:middle line:84% So another way to think about-- so if you go back to what 00:42:35.350 --> 00:42:39.880 align:middle line:84% eigenvalues, eigenvectors were, it's especially clean if you 00:42:39.880 --> 00:42:42.870 align:middle line:90% have what were-- 00:42:42.870 --> 00:42:46.320 align:middle line:84% depending on if you did stuff with complex value 00:42:46.320 --> 00:42:52.350 align:middle line:84% matrices or not, but let's say they're symmetric, 00:42:52.350 --> 00:42:54.460 align:middle line:84% and you're dealing with real value matrices, 00:42:54.460 --> 00:42:56.850 align:middle line:84% so matrices with real entries, then 00:42:56.850 --> 00:43:02.220 align:middle line:84% a theorem of linear algebra is that you can diagonalize 00:43:02.220 --> 00:43:06.240 align:middle line:84% that matrix, meaning you can find an orthonormal basis 00:43:06.240 --> 00:43:13.470 align:middle line:84% of R n So that in that basis, the matrix is simply diagonal, 00:43:13.470 --> 00:43:15.420 align:middle line:84% and the numbers appearing on the diagonal 00:43:15.420 --> 00:43:19.110 align:middle line:84% are the eigenvalues of that operator. 00:43:19.110 --> 00:43:26.980 align:middle line:84% Now, another way to think about that is that for which lambda, 00:43:26.980 --> 00:43:30.820 align:middle line:84% say, can I invert the matrix A minus lambda? 00:43:30.820 --> 00:43:36.430 align:middle line:84% In fact, that's exactly how the eigenvalues are defined. 00:43:36.430 --> 00:43:37.930 align:middle line:84% If you want to find an eigenvector, 00:43:37.930 --> 00:43:41.390 align:middle line:84% then that should be in the null space of A minus lambda. 00:43:41.390 --> 00:43:43.090 align:middle line:84% So you compute the determinant of that, 00:43:43.090 --> 00:43:45.622 align:middle line:84% and you get a polynomial that you saw for the lambdas. 00:43:45.622 --> 00:43:47.830 align:middle line:84% And then you get to solve for the eigenvectors, which 00:43:47.830 --> 00:43:51.430 align:middle line:84% is a system of linear equations once you know lambda. 00:43:51.430 --> 00:43:57.280 align:middle line:84% But thinking back, those eigenvalues 00:43:57.280 --> 00:44:06.980 align:middle line:84% are impediments to being able to solve the equation A minus 00:44:06.980 --> 00:44:09.080 align:middle line:90% lambda u equals f. 00:44:09.080 --> 00:44:13.670 align:middle line:84% And so with that in mind, that's how 00:44:13.670 --> 00:44:18.210 align:middle line:84% we'll define the spectrum of a bounded linear operator. 00:44:18.210 --> 00:44:23.130 align:middle line:84% So first, I'm going to define the complement of the spectrum. 00:44:23.130 --> 00:44:26.840 align:middle line:84% So let A be a bounded linear operator. 00:44:26.840 --> 00:44:31.490 align:middle line:90% 00:44:31.490 --> 00:44:32.930 align:middle line:90% The resolvent set of A-- 00:44:32.930 --> 00:44:38.310 align:middle line:90% 00:44:38.310 --> 00:44:39.570 align:middle line:90% so this is the new bit-- 00:44:39.570 --> 00:44:43.560 align:middle line:90% 00:44:43.560 --> 00:44:53.970 align:middle line:84% is a set Res A. This consists of all complex numbers lambda, 00:44:53.970 --> 00:44:58.290 align:middle line:84% with the property that A minus lambda 00:44:58.290 --> 00:45:03.630 align:middle line:90% I is an invertible operator. 00:45:03.630 --> 00:45:07.460 align:middle line:90% 00:45:07.460 --> 00:45:15.490 align:middle line:84% Now, I'm a bit lazy, and so throughout, 00:45:15.490 --> 00:45:17.680 align:middle line:84% instead of writing A minus lambda I, 00:45:17.680 --> 00:45:20.025 align:middle line:84% will most often write A minus lambda, 00:45:20.025 --> 00:45:21.400 align:middle line:84% with the understanding that there 00:45:21.400 --> 00:45:24.960 align:middle line:90% should be an identity here. 00:45:24.960 --> 00:45:30.030 align:middle line:84% So the resolvent set is a set of all lambdas complex numbers 00:45:30.030 --> 00:45:31.730 align:middle line:84% so that A minus lambda is invertible. 00:45:31.730 --> 00:45:33.188 align:middle line:84% In other words, that you can always 00:45:33.188 --> 00:45:35.180 align:middle line:84% solve the equation A minus lambda 00:45:35.180 --> 00:45:41.190 align:middle line:90% u equals f for arbitrary f. 00:45:41.190 --> 00:45:47.780 align:middle line:84% And you can solve it uniquely, uniquely being the key. 00:45:47.780 --> 00:45:57.570 align:middle line:90% The spectrum of A-- 00:45:57.570 --> 00:46:00.600 align:middle line:84% so this is a new bit of terminology-- 00:46:00.600 --> 00:46:02.220 align:middle line:90% is simply the complement. 00:46:02.220 --> 00:46:09.440 align:middle line:90% 00:46:09.440 --> 00:46:15.110 align:middle line:84% So usually, we write it Spec A equals C take away 00:46:15.110 --> 00:46:31.850 align:middle line:84% the resolvent set of A. So this is the example 00:46:31.850 --> 00:46:33.830 align:middle line:90% I was saying a minute ago. 00:46:33.830 --> 00:46:43.840 align:middle line:84% Let's say I have just a matrix, linear transformation 00:46:43.840 --> 00:46:48.730 align:middle line:84% of C2 to C2, and A is just given by lambda 1, 0, 00:46:48.730 --> 00:46:53.050 align:middle line:84% 0, lambda 2, for some two numbers lambda 1 and lambda 2. 00:46:53.050 --> 00:46:56.880 align:middle line:90% 00:46:56.880 --> 00:46:59.160 align:middle line:84% So I'm giving you the simplest example possible. 00:46:59.160 --> 00:47:04.410 align:middle line:84% Then A minus lambda is equal to lambda 1 minus lambda 0, 0, 00:47:04.410 --> 00:47:06.210 align:middle line:90% lambda 2 minus lambda. 00:47:06.210 --> 00:47:09.210 align:middle line:84% And when is this matrix invertible? 00:47:09.210 --> 00:47:13.380 align:middle line:84% Precisely when lambda does not equal lambda 1 or lambda 2. 00:47:13.380 --> 00:47:20.450 align:middle line:90% 00:47:20.450 --> 00:47:26.570 align:middle line:84% So A minus lambda is invertible if and only 00:47:26.570 --> 00:47:33.650 align:middle line:84% if lambda does not equal lambda 1 or lambda 2. 00:47:33.650 --> 00:47:38.450 align:middle line:84% And when lambda equals lambda 2 or lambda 1, 00:47:38.450 --> 00:47:43.250 align:middle line:84% this matrix here has a non-trivial kernel. 00:47:43.250 --> 00:47:45.680 align:middle line:84% And that's really the only obstruction 00:47:45.680 --> 00:47:49.820 align:middle line:90% to being invertible. 00:47:49.820 --> 00:47:51.950 align:middle line:84% But in infinite dimensions, when we 00:47:51.950 --> 00:47:53.780 align:middle line:84% have operators on infinite dimensions, 00:47:53.780 --> 00:47:59.030 align:middle line:84% there's an additional wrinkle that could happen. 00:47:59.030 --> 00:48:01.310 align:middle line:84% So let me finish with this example. 00:48:01.310 --> 00:48:08.090 align:middle line:84% Therefore, the residual set of A or residual resolvent set 00:48:08.090 --> 00:48:14.600 align:middle line:84% of A equals C take away lambda 1, lambda 2. 00:48:14.600 --> 00:48:26.150 align:middle line:84% And the spectrum of a equals C set lambda 1, lambda 2. 00:48:26.150 --> 00:48:34.750 align:middle line:90% 00:48:34.750 --> 00:48:36.670 align:middle line:84% So we kind of have a special name 00:48:36.670 --> 00:48:43.600 align:middle line:84% for if something pops up in the spectrum, in kind 00:48:43.600 --> 00:48:46.280 align:middle line:84% of the same way we have for regular matrices. 00:48:46.280 --> 00:49:03.180 align:middle line:84% So definition-- if A is a bounded linear operator and A 00:49:03.180 --> 00:49:05.350 align:middle line:90% minus lambda-- 00:49:05.350 --> 00:49:06.870 align:middle line:90% so lambda is a complex number-- 00:49:06.870 --> 00:49:15.540 align:middle line:84% is not injective-- so we need two conditions for A minus 00:49:15.540 --> 00:49:17.710 align:middle line:84% lambda, or for lambda to be in the resolvent set, 00:49:17.710 --> 00:49:22.380 align:middle line:84% we need A minus lambda to be both injective and surjective, 00:49:22.380 --> 00:49:25.170 align:middle line:90% one to one and onto. 00:49:25.170 --> 00:49:28.005 align:middle line:84% Then this implies that it has non-trivial kernel. 00:49:28.005 --> 00:49:34.850 align:middle line:90% 00:49:34.850 --> 00:49:39.680 align:middle line:84% And there exists a u in H, take away 0, 00:49:39.680 --> 00:49:44.030 align:middle line:90% so that A u equals lambda u. 00:49:44.030 --> 00:49:47.770 align:middle line:90% 00:49:47.770 --> 00:49:54.040 align:middle line:84% We then call this lambda, which is in the spectrum of A-- 00:49:54.040 --> 00:49:57.340 align:middle line:84% it's in the spectrum because A minus lambda is not 00:49:57.340 --> 00:49:58.120 align:middle line:90% invertible-- 00:49:58.120 --> 00:50:01.640 align:middle line:90% 00:50:01.640 --> 00:50:17.450 align:middle line:84% an eigenvalue of A, and u an eigenvector. 00:50:17.450 --> 00:50:25.110 align:middle line:90% 00:50:25.110 --> 00:50:31.000 align:middle line:84% So the best way to learn about spectrum 00:50:31.000 --> 00:50:34.100 align:middle line:90% is to do a lot of examples. 00:50:34.100 --> 00:50:37.390 align:middle line:84% And so I only have a finite amount of time in class 00:50:37.390 --> 00:50:40.240 align:middle line:84% to teach you new stuff, so I can't do too many examples. 00:50:40.240 --> 00:50:46.830 align:middle line:84% But let's take, for example, the operator from earlier, 00:50:46.830 --> 00:50:58.690 align:middle line:84% T a equals a1 over 1, a2 over 2, a3 over 3, and so on, 00:50:58.690 --> 00:51:03.100 align:middle line:84% for an element of little l2, so a sequence in little L2. 00:51:03.100 --> 00:51:05.950 align:middle line:84% And so this was, in fact, a compact operator. 00:51:05.950 --> 00:51:09.740 align:middle line:90% 00:51:09.740 --> 00:51:11.890 align:middle line:84% What would be some eigenvalues of this? 00:51:11.890 --> 00:51:17.760 align:middle line:84% Well, note that if I apply T to one 00:51:17.760 --> 00:51:20.980 align:middle line:84% of the elements of orthonormal basis-- 00:51:20.980 --> 00:51:28.890 align:middle line:84% so let's call it e n, where e n is 00:51:28.890 --> 00:51:34.650 align:middle line:84% the sequence consisting of 1 in the n-th spot and 0 otherwise. 00:51:34.650 --> 00:51:35.590 align:middle line:90% What do I get? 00:51:35.590 --> 00:51:41.250 align:middle line:84% I get 0, 0, except in the n-th spot, 1 over n, 00:51:41.250 --> 00:51:48.270 align:middle line:84% 0, and so on, which is 1 over n times e n. 00:51:48.270 --> 00:52:01.470 align:middle line:84% So that proves that T minus 1 over n has non-trivial kernel, 00:52:01.470 --> 00:52:06.448 align:middle line:84% and therefore, 1 over n is an eigenvalue, 00:52:06.448 --> 00:52:07.740 align:middle line:90% and therefore, in the spectrum. 00:52:07.740 --> 00:52:11.910 align:middle line:90% 00:52:11.910 --> 00:52:13.215 align:middle line:90% So this was for all n. 00:52:13.215 --> 00:52:17.970 align:middle line:90% 00:52:17.970 --> 00:52:25.650 align:middle line:90% These are eigenvalues of T. 00:52:25.650 --> 00:52:46.930 align:middle line:84% So the spectrum contains at least this sequence of numbers. 00:52:46.930 --> 00:52:49.080 align:middle line:84% So unlike in the finite-dimensional case, 00:52:49.080 --> 00:52:51.930 align:middle line:84% you can have infinitely many eigenvalues. 00:52:51.930 --> 00:52:55.580 align:middle line:90% 00:52:55.580 --> 00:52:58.750 align:middle line:84% So it's also possible that you have no eigenvalues. 00:52:58.750 --> 00:53:08.750 align:middle line:90% 00:53:08.750 --> 00:53:12.000 align:middle line:84% Let me just bring up a small point here. 00:53:12.000 --> 00:53:16.840 align:middle line:90% 00:53:16.840 --> 00:53:19.000 align:middle line:90% Let's look at 0. 00:53:19.000 --> 00:53:24.280 align:middle line:84% Is 0 in the spectrum or in the resultant? 00:53:24.280 --> 00:53:28.240 align:middle line:84% In other words, So T minus 0, that's just T. So I'm asking 00:53:28.240 --> 00:53:35.500 align:middle line:84% is T itself injective and surjective? 00:53:35.500 --> 00:53:37.090 align:middle line:90% Is it bijective? 00:53:37.090 --> 00:53:39.790 align:middle line:84% So first off, it should be clear that this operator here 00:53:39.790 --> 00:53:40.990 align:middle line:90% is injective. 00:53:40.990 --> 00:53:46.450 align:middle line:84% If T a equals 0, then every one of these numbers is 0, 00:53:46.450 --> 00:53:48.160 align:middle line:90% and therefore, a has to be 0. 00:53:48.160 --> 00:53:49.870 align:middle line:90% So T is clearly injective. 00:53:49.870 --> 00:53:52.990 align:middle line:90% 00:53:52.990 --> 00:53:59.310 align:middle line:84% But try to convince yourself that T cannot be surjective, 00:53:59.310 --> 00:54:01.860 align:middle line:84% because if it was surjective, the inverse would have to be 00:54:01.860 --> 00:54:03.720 align:middle line:90% a bounded linear operator. 00:54:03.720 --> 00:54:05.340 align:middle line:84% What would the inverse have to be? 00:54:05.340 --> 00:54:10.740 align:middle line:84% It would have to be I take the elements of my sequence in l2, 00:54:10.740 --> 00:54:16.470 align:middle line:84% and a1 gets multiplied by 1, a2 gets multiplied by 2, 00:54:16.470 --> 00:54:20.430 align:middle line:84% a3 gets multiplied by 3, and so on. 00:54:20.430 --> 00:54:24.750 align:middle line:84% And that would not be a bounded linear operator on little l2. 00:54:24.750 --> 00:54:31.140 align:middle line:84% So that argument tells you that it can't be surjective. 00:54:31.140 --> 00:54:36.030 align:middle line:84% But what one can show is that the range is dense, 00:54:36.030 --> 00:54:37.020 align:middle line:90% though, in little l2. 00:54:37.020 --> 00:54:38.730 align:middle line:90% So you can have this-- 00:54:38.730 --> 00:54:41.718 align:middle line:84% something can be in the spectrum, something 00:54:41.718 --> 00:54:43.260 align:middle line:84% that's completely different than what 00:54:43.260 --> 00:54:45.670 align:middle line:90% happens in finite dimensions. 00:54:45.670 --> 00:54:53.090 align:middle line:84% You can have the range of T or T minus lambda, 00:54:53.090 --> 00:54:59.330 align:middle line:84% say-- you could have the range be dense in H, but not closed. 00:54:59.330 --> 00:55:02.510 align:middle line:84% And therefore, the operator is not surjective. 00:55:02.510 --> 00:55:06.740 align:middle line:84% This one subtle difference between what 00:55:06.740 --> 00:55:09.440 align:middle line:84% happens in infinite dimensions and finite dimensions-- that 00:55:09.440 --> 00:55:12.680 align:middle line:84% doesn't happen in finite dimensions because the range 00:55:12.680 --> 00:55:16.100 align:middle line:84% is going to be a subspace of H, and in finite dimensions, 00:55:16.100 --> 00:55:18.750 align:middle line:84% the range is a finite-dimensional subspace, 00:55:18.750 --> 00:55:20.480 align:middle line:90% and so it's always closed. 00:55:20.480 --> 00:55:21.950 align:middle line:84% In infinite dimensions, the range 00:55:21.950 --> 00:55:23.240 align:middle line:90% does not have to be closed. 00:55:23.240 --> 00:55:25.430 align:middle line:84% So this little difference, though, 00:55:25.430 --> 00:55:29.150 align:middle line:84% can result in something being in the spectrum. 00:55:29.150 --> 00:55:34.220 align:middle line:84% It's not always just something's in the spectrum if 00:55:34.220 --> 00:55:35.540 align:middle line:90% and only if it's an eigenvalue. 00:55:35.540 --> 00:55:37.380 align:middle line:90% That is not the case. 00:55:37.380 --> 00:55:42.800 align:middle line:84% And so for example, and I'll probably 00:55:42.800 --> 00:55:57.290 align:middle line:84% put this on the assignment, but what you can verify 00:55:57.290 --> 00:56:01.700 align:middle line:84% is that let's say I define now an operator T from L2 of 0, 00:56:01.700 --> 00:56:14.940 align:middle line:84% 1 to L2 0, 1 by T f of x equals x times f of x. 00:56:14.940 --> 00:56:20.977 align:middle line:90% 00:56:20.977 --> 00:56:33.990 align:middle line:84% Then T has no, none, eigenvalues. 00:56:33.990 --> 00:56:36.590 align:middle line:84% So I'll probably put this example either on the homework 00:56:36.590 --> 00:56:39.930 align:middle line:84% or on the exam, but it has no eigenvalues. 00:56:39.930 --> 00:56:43.490 align:middle line:84% And you can compute that the spectrum, 00:56:43.490 --> 00:56:50.360 align:middle line:84% the set of all lambda so that T minus lambda is not invertible, 00:56:50.360 --> 00:56:51.380 align:middle line:90% equals 0, 1. 00:56:51.380 --> 00:56:54.320 align:middle line:90% 00:56:54.320 --> 00:56:55.220 align:middle line:90% Why 0, 1? 00:56:55.220 --> 00:56:56.780 align:middle line:84% It's not because the domain's 0, 1, 00:56:56.780 --> 00:57:00.680 align:middle line:84% but because that's essentially the range of x, the function 00:57:00.680 --> 00:57:03.260 align:middle line:90% x as x ranges from 0 to 1. 00:57:03.260 --> 00:57:06.110 align:middle line:90% 00:57:06.110 --> 00:57:09.050 align:middle line:84% So when you move on to infinite dimensions, 00:57:09.050 --> 00:57:10.610 align:middle line:84% the spectrum doesn't necessarily need 00:57:10.610 --> 00:57:13.760 align:middle line:84% to just be the eigenvalues of your operator. 00:57:13.760 --> 00:57:17.000 align:middle line:84% In fact, there may be no eigenvalues. 00:57:17.000 --> 00:57:19.200 align:middle line:84% It can be something that's much more subtle, 00:57:19.200 --> 00:57:24.290 align:middle line:84% which occurs when the range is not a closed subset, is not a-- 00:57:24.290 --> 00:57:27.780 align:middle line:90% 00:57:27.780 --> 00:57:31.200 align:middle line:84% the range can be dense in H, but not close, and therefore, 00:57:31.200 --> 00:57:34.080 align:middle line:84% not equal to all of H. So the operator is not surjective. 00:57:34.080 --> 00:57:43.620 align:middle line:90% 00:57:43.620 --> 00:57:50.510 align:middle line:84% So let's prove some general properties 00:57:50.510 --> 00:57:52.970 align:middle line:90% of spectrum and resolvent sets. 00:57:52.970 --> 00:57:59.370 align:middle line:84% So let A be a bounded linear operator. 00:57:59.370 --> 00:58:02.520 align:middle line:90% 00:58:02.520 --> 00:58:18.560 align:middle line:84% Then spectrum of A is a closed subset of C. 00:58:18.560 --> 00:58:27.280 align:middle line:84% And in fact, it's contained in the set of lambdas 00:58:27.280 --> 00:58:39.060 align:middle line:84% in C with modulus less than or equal to the norm of A. 00:58:39.060 --> 00:58:42.000 align:middle line:84% So in particular, this implies that the spectrum 00:58:42.000 --> 00:58:44.250 align:middle line:84% is a compact subset of the complex numbers. 00:58:44.250 --> 00:58:46.960 align:middle line:90% 00:58:46.960 --> 00:58:50.280 align:middle line:84% So in particular, since 1 over n is 00:58:50.280 --> 00:58:52.950 align:middle line:84% in the spectrum of this operator we have over here, 00:58:52.950 --> 00:58:55.860 align:middle line:84% and 0 is the limit as n goes to infinity of 1 over n, 00:58:55.860 --> 00:58:57.450 align:middle line:84% this automatically, from this result, 00:58:57.450 --> 00:59:01.110 align:middle line:84% implies 0 is in the spectrum of this operator. 00:59:01.110 --> 00:59:03.680 align:middle line:90% 00:59:03.680 --> 00:59:08.090 align:middle line:84% Now of course, you can phrase this, since the spectrum of A 00:59:08.090 --> 00:59:11.880 align:middle line:84% is the complement of the resolvent set of A, 00:59:11.880 --> 00:59:13.640 align:middle line:84% you could have phrased this theorem 00:59:13.640 --> 00:59:16.430 align:middle line:84% in terms of the resolvent, which is that the resolvent set of A 00:59:16.430 --> 00:59:19.670 align:middle line:84% is open and the resolvent set of A 00:59:19.670 --> 00:59:21.560 align:middle line:84% contains the complement of this set. 00:59:21.560 --> 00:59:25.630 align:middle line:84% That's, in fact, how we'll prove this theorem. 00:59:25.630 --> 00:59:38.630 align:middle line:84% We'll show-- is open and that this thing, the complement 00:59:38.630 --> 00:59:41.150 align:middle line:84% of this set, is contained in the resolvent of A, 00:59:41.150 --> 00:59:49.940 align:middle line:84% and set of lambdas with modulus bigger than the norm of A 00:59:49.940 --> 01:00:11.960 align:middle line:84% is contained in the resolvent set of A. 01:00:11.960 --> 01:00:13.430 align:middle line:90% So why is the resolvent open? 01:00:13.430 --> 01:00:17.390 align:middle line:84% Well, it follows from the fact that GL H, 01:00:17.390 --> 01:00:21.290 align:middle line:84% the space of bounded linear operators that are invertible, 01:00:21.290 --> 01:00:22.490 align:middle line:90% is an open set. 01:00:22.490 --> 01:00:28.130 align:middle line:84% So take something in the resolvent. 01:00:28.130 --> 01:00:30.200 align:middle line:84% We have to show that there's a small ball 01:00:30.200 --> 01:00:34.647 align:middle line:84% around this complex number so that that small ball 01:00:34.647 --> 01:00:36.230 align:middle line:84% is contained in the resolvent, meaning 01:00:36.230 --> 01:00:39.560 align:middle line:90% A minus lambda is invertible. 01:00:39.560 --> 01:00:42.670 align:middle line:90% 01:00:42.670 --> 01:00:57.340 align:middle line:84% Since GL H is open, this implies there exists some small epsilon 01:00:57.340 --> 01:01:01.780 align:middle line:84% such that if T is an operator which 01:01:01.780 --> 01:01:09.220 align:middle line:84% is close to A minus lambda 0, an operator norm, 01:01:09.220 --> 01:01:16.300 align:middle line:84% this implies that T is also invertible. 01:01:16.300 --> 01:01:22.870 align:middle line:90% 01:01:22.870 --> 01:01:30.040 align:middle line:84% Then, if lambda minus lambda 0 is less than epsilon, 01:01:30.040 --> 01:01:36.940 align:middle line:84% we get that the norm of A minus lambda minus A0 minus lambda 01:01:36.940 --> 01:01:43.000 align:middle line:84% 0, the A's cancel and I just get the operator 01:01:43.000 --> 01:01:47.380 align:middle line:90% norm of lambda minus lambda 0-- 01:01:47.380 --> 01:01:52.000 align:middle line:84% and here, I should have kept around the identities-- 01:01:52.000 --> 01:01:56.200 align:middle line:84% which just equals lambda minus lambda 0, which we're assuming 01:01:56.200 --> 01:01:57.530 align:middle line:90% is less than epsilon. 01:01:57.530 --> 01:02:01.660 align:middle line:84% So this operator here is within epsilon 01:02:01.660 --> 01:02:04.100 align:middle line:84% of this convertible operator here, 01:02:04.100 --> 01:02:13.550 align:middle line:84% which guarantees, by how we chose epsilon, 01:02:13.550 --> 01:02:17.990 align:middle line:84% that A minus lambda is invertible, 01:02:17.990 --> 01:02:24.950 align:middle line:84% and therefore, lambda is in the resolvent set of A. 01:02:24.950 --> 01:02:31.040 align:middle line:84% So we started out with something within epsilon distance 01:02:31.040 --> 01:02:34.790 align:middle line:84% to lambda 0, where epsilon was coming from this condition 01:02:34.790 --> 01:02:42.020 align:middle line:84% here, and showed that lambda is in the resolvent, i.e., 01:02:42.020 --> 01:02:44.420 align:middle line:84% the ball lambda minus lambda 0 less 01:02:44.420 --> 01:02:50.450 align:middle line:84% than epsilon is contained in the resolvent set of A. 01:02:50.450 --> 01:02:53.870 align:middle line:84% So that proves that the resolvent set is open. 01:02:53.870 --> 01:03:00.450 align:middle line:90% 01:03:00.450 --> 01:03:03.420 align:middle line:84% So now, let's show this condition-- 01:03:03.420 --> 01:03:05.580 align:middle line:84% that if the absolute value of lambda 01:03:05.580 --> 01:03:11.310 align:middle line:84% is bigger than the norm of A, then A minus lambda 01:03:11.310 --> 01:03:12.270 align:middle line:90% is invertible. 01:03:12.270 --> 01:03:15.560 align:middle line:90% 01:03:15.560 --> 01:03:19.790 align:middle line:84% So suppose lambda is bigger than, strictly 01:03:19.790 --> 01:03:25.300 align:middle line:84% bigger than, the norm of A. So in particular, it's nonzero. 01:03:25.300 --> 01:03:37.740 align:middle line:84% Then the norm of 1 over lambda times A is less than 1, 01:03:37.740 --> 01:03:53.710 align:middle line:84% and this implies that I minus 1 over lambda A is invertible. 01:03:53.710 --> 01:03:55.660 align:middle line:84% Because then this has norm less than 1. 01:03:55.660 --> 01:03:57.220 align:middle line:84% So we can always invert something 01:03:57.220 --> 01:04:01.960 align:middle line:84% that is I minus that thing that has norm less than 1, 01:04:01.960 --> 01:04:06.940 align:middle line:84% which tells me that A minus lambda is the product of two 01:04:06.940 --> 01:04:13.120 align:middle line:84% invertible operators, namely multiplication by minus lambda 01:04:13.120 --> 01:04:20.780 align:middle line:90% and I minus 1 over lambda A. 01:04:20.780 --> 01:04:21.280 align:middle line:90% Let's see. 01:04:21.280 --> 01:04:22.738 align:middle line:84% Let's make sure everything cancels. 01:04:22.738 --> 01:04:24.500 align:middle line:90% OK. 01:04:24.500 --> 01:04:27.110 align:middle line:84% So multiplication by lambda, which 01:04:27.110 --> 01:04:30.320 align:middle line:84% is a positive or nonzero complex number, that's 01:04:30.320 --> 01:04:31.790 align:middle line:90% an invertible operator. 01:04:31.790 --> 01:04:33.770 align:middle line:90% This is an invertible operator. 01:04:33.770 --> 01:04:35.540 align:middle line:84% Product of two invertible operators 01:04:35.540 --> 01:04:41.330 align:middle line:84% is invertible, so that thing is in GL H. 01:04:41.330 --> 01:04:46.970 align:middle line:84% So it's invertible, which implies lambda, simply 01:04:46.970 --> 01:04:50.850 align:middle line:84% by definition, is in the resolvent set of A. 01:04:50.850 --> 01:04:54.860 align:middle line:84% So we supposed that we had something here, and showed 01:04:54.860 --> 01:04:57.890 align:middle line:84% it's in here, which is what we wanted to do. 01:04:57.890 --> 01:05:00.990 align:middle line:90% 01:05:00.990 --> 01:05:09.330 align:middle line:84% So the spectrum is a closed subset, 01:05:09.330 --> 01:05:13.635 align:middle line:84% in fact, a compact subset, of a set of complex numbers. 01:05:13.635 --> 01:05:16.440 align:middle line:90% 01:05:16.440 --> 01:05:18.630 align:middle line:84% At least for this class, that's about as much 01:05:18.630 --> 01:05:26.070 align:middle line:84% as we can say for general bounded linear operators 01:05:26.070 --> 01:05:27.840 align:middle line:90% about their spectrum. 01:05:27.840 --> 01:05:30.270 align:middle line:84% Let me just make a small comment, also, 01:05:30.270 --> 01:05:37.800 align:middle line:84% that if you know a little complex analysis, 01:05:37.800 --> 01:05:43.080 align:middle line:84% the previous proof actually also allows you to-- 01:05:43.080 --> 01:05:45.190 align:middle line:84% so I'm just going to state this as a fact. 01:05:45.190 --> 01:05:49.330 align:middle line:84% One could ask, could the spectrum ever be empty? 01:05:49.330 --> 01:05:53.940 align:middle line:84% So that's certainly a compact subset of complex numbers. 01:05:53.940 --> 01:05:56.170 align:middle line:90% And the answer is no. 01:05:56.170 --> 01:05:59.020 align:middle line:84% But this class does not assume complex analysis, 01:05:59.020 --> 01:06:02.400 align:middle line:84% so I can't really or write a proof of that 01:06:02.400 --> 01:06:04.810 align:middle line:84% and expect you to completely follow it. 01:06:04.810 --> 01:06:07.110 align:middle line:90% But here's the thinking-- 01:06:07.110 --> 01:06:08.880 align:middle line:84% so if you don't know complex analysis, 01:06:08.880 --> 01:06:12.360 align:middle line:84% just skip ahead a few seconds, but the thinking 01:06:12.360 --> 01:06:17.050 align:middle line:84% is this-- let's suppose that the spectrum is empty. 01:06:17.050 --> 01:06:27.180 align:middle line:90% Then A minus lambda applied-- 01:06:27.180 --> 01:06:31.190 align:middle line:84% so let's assume the spectrum is empty, and take u and v that 01:06:31.190 --> 01:06:36.230 align:middle line:84% are in the Hilbert space, apply A minus lambda to u or A minus 01:06:36.230 --> 01:06:39.170 align:middle line:90% lambda inverse, applied to u. 01:06:39.170 --> 01:06:40.850 align:middle line:90% An inner product it would be. 01:06:40.850 --> 01:06:44.750 align:middle line:84% So this is well defined because A minus lambda inverse always 01:06:44.750 --> 01:06:48.560 align:middle line:84% exists because the spectrum is empty. 01:06:48.560 --> 01:06:51.230 align:middle line:84% And this gives you a continuous function 01:06:51.230 --> 01:06:55.160 align:middle line:90% in lambda on the complex plane. 01:06:55.160 --> 01:06:56.790 align:middle line:90% And now what happens? 01:06:56.790 --> 01:07:00.950 align:middle line:84% So the proof, or at least one of the estimates I gave before, 01:07:00.950 --> 01:07:10.360 align:middle line:84% says that as the magnitude of lambda gets large, 01:07:10.360 --> 01:07:12.460 align:middle line:84% the operator norm of A minus lambda 01:07:12.460 --> 01:07:18.140 align:middle line:84% is bounded by a constant divided by the magnitude of lambda, 01:07:18.140 --> 01:07:19.870 align:middle line:90% which is going to 0. 01:07:19.870 --> 01:07:24.700 align:middle line:84% So the operator norm of the inverse of A minus lambda 01:07:24.700 --> 01:07:31.540 align:middle line:84% goes to 0 as the magnitude of lambda goes to infinity. 01:07:31.540 --> 01:07:36.210 align:middle line:84% But you can show a bit more about this function 01:07:36.210 --> 01:07:39.090 align:middle line:84% I defined a minute ago involving two vectors-- 01:07:39.090 --> 01:07:42.570 align:middle line:84% that basically, by this Neumann series calculation 01:07:42.570 --> 01:07:44.910 align:middle line:84% that in fact, that's a complex differentiable 01:07:44.910 --> 01:07:47.630 align:middle line:90% function in lambda. 01:07:47.630 --> 01:07:51.020 align:middle line:84% And thus, you have this complex differentiable function 01:07:51.020 --> 01:07:57.040 align:middle line:84% defined on the entire plane that goes to 0 as lambda 01:07:57.040 --> 01:07:59.480 align:middle line:90% goes to infinity. 01:07:59.480 --> 01:08:01.480 align:middle line:90% And by Liouville's theorem-- 01:08:01.480 --> 01:08:03.010 align:middle line:84% not Louisville, Louisville's a type 01:08:03.010 --> 01:08:07.850 align:middle line:84% of bat, a bat that you hit stuff with, 01:08:07.850 --> 01:08:11.310 align:middle line:84% not the bat that gives you COVID-- 01:08:11.310 --> 01:08:15.120 align:middle line:84% then A minus lambda inverse applied to u inner product v 01:08:15.120 --> 01:08:18.359 align:middle line:84% has to be identically 0 because that's a complex differentiable 01:08:18.359 --> 01:08:20.609 align:middle line:84% function in lambda, which goes to 0 as lambda 01:08:20.609 --> 01:08:22.000 align:middle line:90% goes to infinity. 01:08:22.000 --> 01:08:23.910 align:middle line:84% And since this applies for all u and v, 01:08:23.910 --> 01:08:27.359 align:middle line:84% this implies that the operator A minus lambda inverse 01:08:27.359 --> 01:08:28.350 align:middle line:90% is identically 0. 01:08:28.350 --> 01:08:29.529 align:middle line:90% That's a contradiction. 01:08:29.529 --> 01:08:33.680 align:middle line:90% So the spectrum cannot be empty. 01:08:33.680 --> 01:08:41.850 align:middle line:84% So it will be very useful to have 01:08:41.850 --> 01:08:43.439 align:middle line:90% a different characterization. 01:08:43.439 --> 01:08:47.490 align:middle line:84% So now, we're going to specialize to not just looking 01:08:47.490 --> 01:08:51.390 align:middle line:84% at the spectrum of arbitrary bounded operators, 01:08:51.390 --> 01:08:55.649 align:middle line:84% but now, we're going to zero in on self-adjoint operators. 01:08:55.649 --> 01:08:58.340 align:middle line:90% 01:08:58.340 --> 01:09:03.290 align:middle line:84% And first, we need a little theorem, 01:09:03.290 --> 01:09:06.859 align:middle line:84% that if I have a self-adjoint operator, 01:09:06.859 --> 01:09:14.510 align:middle line:84% meaning A is equal to A star, then two things are true-- 01:09:14.510 --> 01:09:17.090 align:middle line:90% 01:09:17.090 --> 01:09:25.910 align:middle line:84% for all u in H, the inner product A u applied to u 01:09:25.910 --> 01:09:28.310 align:middle line:90% is a real number. 01:09:28.310 --> 01:09:31.430 align:middle line:84% And two, I can give an alternative characterization 01:09:31.430 --> 01:09:36.800 align:middle line:84% of the norm of A, that the norm of A 01:09:36.800 --> 01:09:40.520 align:middle line:84% is, in fact, equal to the sup over norm of u 01:09:40.520 --> 01:09:46.142 align:middle line:84% equals 1 of the absolute value of A u applied to u. 01:09:46.142 --> 01:09:48.350 align:middle line:84% So this actually has nothing to do with the spectrum. 01:09:48.350 --> 01:09:52.580 align:middle line:84% This is just a fact I'll need when 01:09:52.580 --> 01:09:58.550 align:middle line:84% we start studying the spectrum of self-adjoint operators. 01:09:58.550 --> 01:10:01.020 align:middle line:84% And that will be in the next lecture. 01:10:01.020 --> 01:10:03.380 align:middle line:84% So with the remainder of this lecture, 01:10:03.380 --> 01:10:05.700 align:middle line:90% we'll prove this theorem. 01:10:05.700 --> 01:10:08.960 align:middle line:90% So proof-- so 1 is quite easy. 01:10:08.960 --> 01:10:13.680 align:middle line:90% 01:10:13.680 --> 01:10:18.330 align:middle line:84% If u is in H, and I look at A u applied to u, 01:10:18.330 --> 01:10:20.010 align:middle line:84% and I take its complex conjugate, 01:10:20.010 --> 01:10:22.860 align:middle line:84% something's real if and only if its complex conjugate 01:10:22.860 --> 01:10:24.900 align:middle line:90% is equal to the number again. 01:10:24.900 --> 01:10:27.120 align:middle line:84% So I want to show that the complex conjugate 01:10:27.120 --> 01:10:31.390 align:middle line:84% of A u applied to u is equal to A u applied to u. 01:10:31.390 --> 01:10:34.830 align:middle line:84% So this is equal to, by the properties 01:10:34.830 --> 01:10:39.900 align:middle line:84% of the inner product, u applied to A u. 01:10:39.900 --> 01:10:42.960 align:middle line:84% Now, A is equal to its own adjoint 01:10:42.960 --> 01:10:45.930 align:middle line:84% so this is equal to u applied to-- 01:10:45.930 --> 01:10:48.120 align:middle line:90% or u inner product A star u. 01:10:48.120 --> 01:10:54.020 align:middle line:84% And by the definition of the adjoint, this moves over here. 01:10:54.020 --> 01:10:56.970 align:middle line:90% And that's the end. 01:10:56.970 --> 01:10:59.640 align:middle line:84% Now, I keep bringing up quantum mechanics, 01:10:59.640 --> 01:11:08.460 align:middle line:84% but I am a researcher in quantum mechanics, but I'm not. 01:11:08.460 --> 01:11:14.250 align:middle line:84% But in quantum mechanics, the observables-- 01:11:14.250 --> 01:11:16.470 align:middle line:84% meaning the things that one would actually 01:11:16.470 --> 01:11:21.482 align:middle line:84% measure like position, momentum, center of mass, 01:11:21.482 --> 01:11:22.440 align:middle line:90% these types of things-- 01:11:22.440 --> 01:11:25.070 align:middle line:90% 01:11:25.070 --> 01:11:29.100 align:middle line:84% are modeled by self-adjoint operators. 01:11:29.100 --> 01:11:30.980 align:middle line:84% These are not bounded linear operators. 01:11:30.980 --> 01:11:34.430 align:middle line:84% They're unbounded, which we will not cover in this class, 01:11:34.430 --> 01:11:37.100 align:middle line:90% but they're self adjoint. 01:11:37.100 --> 01:11:40.470 align:middle line:84% And therefore, by this first property this quantity here, 01:11:40.470 --> 01:11:44.040 align:middle line:84% which is the expectation of that measurement-- 01:11:44.040 --> 01:11:45.890 align:middle line:84% so what this number 1 tells you is 01:11:45.890 --> 01:11:48.350 align:middle line:84% that the expectation of this measurement 01:11:48.350 --> 01:11:49.440 align:middle line:90% is always a real number. 01:11:49.440 --> 01:11:53.340 align:middle line:84% It's always something that you can measure in nature. 01:11:53.340 --> 01:11:59.450 align:middle line:84% I mean, as far as I know, the things one measures in nature, 01:11:59.450 --> 01:12:03.890 align:middle line:84% not necessarily uses to model nature, 01:12:03.890 --> 01:12:05.340 align:middle line:90% are not complex numbers. 01:12:05.340 --> 01:12:11.510 align:middle line:84% So number 2, let's verify that I can, 01:12:11.510 --> 01:12:14.690 align:middle line:84% for a self-adjoint operator, write the norm in this way. 01:12:14.690 --> 01:12:23.820 align:middle line:90% 01:12:23.820 --> 01:12:28.365 align:middle line:90% So let little a be this thing. 01:12:28.365 --> 01:12:36.910 align:middle line:90% 01:12:36.910 --> 01:12:39.820 align:middle line:84% First thing to note is actually, this is a finite number, 01:12:39.820 --> 01:12:43.870 align:middle line:84% and it's bounded above by the norm of A. 01:12:43.870 --> 01:12:53.070 align:middle line:84% Note that for all unit length vectors, 01:12:53.070 --> 01:12:57.060 align:middle line:84% the absolute value of the inner product of A u with u 01:12:57.060 --> 01:13:00.210 align:middle line:84% via Cauchy-Schwarz is less than or equal to the norm, 01:13:00.210 --> 01:13:02.730 align:middle line:84% the product of the norms of A u and u. 01:13:02.730 --> 01:13:05.670 align:middle line:84% u has norm 1, so this is less than 01:13:05.670 --> 01:13:07.470 align:middle line:90% or equal to the norm of A u. 01:13:07.470 --> 01:13:09.720 align:middle line:84% And by the definition of the operator norm of A, 01:13:09.720 --> 01:13:18.210 align:middle line:84% that's less than or equal to the operator norm of A. 01:13:18.210 --> 01:13:19.410 align:middle line:90% So this proves two things. 01:13:19.410 --> 01:13:22.830 align:middle line:84% This thing on the right-hand side is a finite number 01:13:22.830 --> 01:13:26.070 align:middle line:84% and it's bounded above by the norm of A. 01:13:26.070 --> 01:13:31.215 align:middle line:84% So little a, the sup, is less than or equal to norm of A. 01:13:31.215 --> 01:13:34.890 align:middle line:84% And what we'll do is prove the opposite inequality. 01:13:34.890 --> 01:13:43.840 align:middle line:90% 01:13:43.840 --> 01:13:47.570 align:middle line:84% And to do that, we'll use this first property, that for all u, 01:13:47.570 --> 01:13:49.600 align:middle line:90% A u applied to u is real. 01:13:49.600 --> 01:14:00.070 align:middle line:84% So I want to somehow estimate the operator 01:14:00.070 --> 01:14:05.530 align:middle line:84% norm of A, which means I need to estimate the norm of A u, where 01:14:05.530 --> 01:14:06.730 align:middle line:90% the norm of u equals 1. 01:14:06.730 --> 01:14:10.300 align:middle line:84% So let's take something in H with norm 1. 01:14:10.300 --> 01:14:13.150 align:middle line:90% 01:14:13.150 --> 01:14:18.370 align:middle line:84% And since the operator norm of A-- 01:14:18.370 --> 01:14:19.570 align:middle line:90% so let me just-- 01:14:19.570 --> 01:14:22.520 align:middle line:90% 01:14:22.520 --> 01:14:32.450 align:middle line:84% now what we want to show is that the norm of A is less than 01:14:32.450 --> 01:14:33.980 align:middle line:84% or equal to this number, little a. 01:14:33.980 --> 01:14:41.480 align:middle line:90% 01:14:41.480 --> 01:14:43.070 align:middle line:84% So let's take something with norm 1. 01:14:43.070 --> 01:14:47.310 align:middle line:90% 01:14:47.310 --> 01:14:48.945 align:middle line:84% And so that when I hit it with A, 01:14:48.945 --> 01:14:52.350 align:middle line:84% I don't get 0, because the operator norm of A 01:14:52.350 --> 01:14:59.370 align:middle line:84% is computed by taking the sup of norms of A u, where 01:14:59.370 --> 01:15:00.780 align:middle line:90% u has unit length. 01:15:00.780 --> 01:15:03.870 align:middle line:84% And I just need to estimate those A u's that are nonzero. 01:15:03.870 --> 01:15:09.360 align:middle line:90% 01:15:09.360 --> 01:15:21.210 align:middle line:84% Let v be the vector A u over the norm of A u. 01:15:21.210 --> 01:15:23.220 align:middle line:90% Then this thing has unit length. 01:15:23.220 --> 01:15:33.060 align:middle line:90% 01:15:33.060 --> 01:15:37.020 align:middle line:84% And so now, if I compute the norm 01:15:37.020 --> 01:15:42.630 align:middle line:84% of A u, which is the thing I want to estimate above by a, 01:15:42.630 --> 01:15:48.710 align:middle line:84% this is equal to A u applied to v, 01:15:48.710 --> 01:15:51.840 align:middle line:84% or A u inner product v, because then I 01:15:51.840 --> 01:15:54.180 align:middle line:84% would get A u inner product A u, which 01:15:54.180 --> 01:15:56.790 align:middle line:84% gives me norm of A u squared divided by the norm of A u. 01:15:56.790 --> 01:15:59.730 align:middle line:90% So I just get A u. 01:15:59.730 --> 01:16:06.270 align:middle line:84% Now, this is a real number because it's 01:16:06.270 --> 01:16:07.110 align:middle line:90% equal to this norm. 01:16:07.110 --> 01:16:12.810 align:middle line:84% So I can throw in a real part and not get into any trouble. 01:16:12.810 --> 01:16:18.365 align:middle line:90% And now, I have a-- 01:16:18.365 --> 01:16:22.072 align:middle line:90% 01:16:22.072 --> 01:16:23.530 align:middle line:84% let's see, what would I call that-- 01:16:23.530 --> 01:16:26.100 align:middle line:90% 01:16:26.100 --> 01:16:27.210 align:middle line:90% polarization identity. 01:16:27.210 --> 01:16:29.910 align:middle line:84% That's the name of that, not the parallelogram law. 01:16:29.910 --> 01:16:33.780 align:middle line:84% But I have a type of polarization identity 01:16:33.780 --> 01:16:35.610 align:middle line:84% for this expression, which you can just 01:16:35.610 --> 01:16:41.340 align:middle line:84% verify in the quiet of your own dwelling 01:16:41.340 --> 01:16:48.840 align:middle line:84% that this thing right here, I can write as 1/4 A u plus v, u 01:16:48.840 --> 01:17:02.640 align:middle line:84% plus v, minus A u minus v, u minus v plus i 01:17:02.640 --> 01:17:16.882 align:middle line:84% times A u plus i v, u plus i v minus A u minus i 01:17:16.882 --> 01:17:34.040 align:middle line:90% v v minus i v. 01:17:34.040 --> 01:17:37.370 align:middle line:90% Now, so A u v-- 01:17:37.370 --> 01:17:39.530 align:middle line:84% this inner product here is equal to this thing, 01:17:39.530 --> 01:17:40.950 align:middle line:90% this expression I have here. 01:17:40.950 --> 01:17:42.630 align:middle line:90% The real part is still there. 01:17:42.630 --> 01:17:45.470 align:middle line:84% Now I'm taking the real part of this number. 01:17:45.470 --> 01:17:48.140 align:middle line:90% Now, this number here is real. 01:17:48.140 --> 01:17:52.100 align:middle line:84% This number here is real by number 1. 01:17:52.100 --> 01:17:55.730 align:middle line:84% And therefore, i times this number is purely imaginary. 01:17:55.730 --> 01:17:58.470 align:middle line:84% So when I take the real part, it goes away. 01:17:58.470 --> 01:18:03.170 align:middle line:84% So this is equal to 1/4 times A u 01:18:03.170 --> 01:18:19.000 align:middle line:84% plus v minus A u minus v, u minus v. 01:18:19.000 --> 01:18:20.260 align:middle line:90% I'm missing something there. 01:18:20.260 --> 01:18:38.430 align:middle line:90% 01:18:38.430 --> 01:18:46.960 align:middle line:84% Now, this is A applied to a vector 01:18:46.960 --> 01:18:48.950 align:middle line:84% inner product with that same vector. 01:18:48.950 --> 01:18:52.930 align:middle line:84% So that's less than or equal to 1/4-- 01:18:52.930 --> 01:18:56.110 align:middle line:84% since little a is a supremum of the absolute values 01:18:56.110 --> 01:18:58.750 align:middle line:84% of these expressions where, as long 01:18:58.750 --> 01:19:00.430 align:middle line:84% as these things have unit length, 01:19:00.430 --> 01:19:05.770 align:middle line:84% this is less than or equal to a times norm of u plus v 01:19:05.770 --> 01:19:08.260 align:middle line:90% squared, this first expression. 01:19:08.260 --> 01:19:10.990 align:middle line:84% Because this is less than or equal to the absolute value 01:19:10.990 --> 01:19:12.160 align:middle line:90% of this thing. 01:19:12.160 --> 01:19:16.000 align:middle line:84% And dividing by u plus v squared, this, 01:19:16.000 --> 01:19:18.040 align:middle line:84% I get that that's less than or equal to a. 01:19:18.040 --> 01:19:21.010 align:middle line:84% But I still have this u plus v squared hanging around. 01:19:21.010 --> 01:19:23.050 align:middle line:84% And then the same thing with this u minus v, 01:19:23.050 --> 01:19:26.830 align:middle line:90% so plus a u minus v squared. 01:19:26.830 --> 01:19:29.930 align:middle line:90% 01:19:29.930 --> 01:19:32.800 align:middle line:84% And now, I use the parallelogram law. 01:19:32.800 --> 01:19:36.790 align:middle line:84% This is equal to a over 4, u plus v norm 01:19:36.790 --> 01:19:39.640 align:middle line:84% squared plus u minus v norm squared. 01:19:39.640 --> 01:19:44.680 align:middle line:84% This is equal to 2 norm u squared plus 2 norm v 01:19:44.680 --> 01:19:47.950 align:middle line:90% squared times a over 4. 01:19:47.950 --> 01:19:51.020 align:middle line:90% u and v have unit length. 01:19:51.020 --> 01:19:52.870 align:middle line:90% So this is 1, this is 1. 01:19:52.870 --> 01:19:55.250 align:middle line:90% So I get 2 plus 2 is 4. 01:19:55.250 --> 01:19:58.470 align:middle line:90% And I get a 01:19:58.470 --> 01:20:05.800 align:middle line:84% So thus, for all norm 1 vectors, I've 01:20:05.800 --> 01:20:11.380 align:middle line:84% proven that the norm of A u is less than or equal to a. 01:20:11.380 --> 01:20:13.870 align:middle line:84% I've done it just for those so that A u is nonzero, 01:20:13.870 --> 01:20:16.660 align:middle line:84% but I still have this inequality of A u equals 0. 01:20:16.660 --> 01:20:21.940 align:middle line:84% So I get that the norm of A is less than or equal to a, 01:20:21.940 --> 01:20:24.460 align:middle line:90% which is what I wanted to do. 01:20:24.460 --> 01:20:35.750 align:middle line:84% So next time, we will get into some properties 01:20:35.750 --> 01:20:47.880 align:middle line:84% of the spectrum of self-adjoint operators a little more, 01:20:47.880 --> 01:20:51.930 align:middle line:84% in particular, the type of subset of complex numbers 01:20:51.930 --> 01:20:52.500 align:middle line:90% it has to be. 01:20:52.500 --> 01:20:55.830 align:middle line:84% In fact, we'll find out that the spectrum of a self-adjoint 01:20:55.830 --> 01:20:59.860 align:middle line:84% operator has to be contained in the real line-- 01:20:59.860 --> 01:21:04.950 align:middle line:84% so you can't have anything with a nonzero imaginary part-- 01:21:04.950 --> 01:21:10.920 align:middle line:84% that's contained within certain bounds related 01:21:10.920 --> 01:21:16.350 align:middle line:84% to this type of expression, A u applied to u. 01:21:16.350 --> 01:21:22.080 align:middle line:84% And this also gives us an interesting way 01:21:22.080 --> 01:21:26.010 align:middle line:84% of checking whether the spectrum is contained 01:21:26.010 --> 01:21:29.887 align:middle line:90% within two real numbers. 01:21:29.887 --> 01:21:31.470 align:middle line:84% In particular, it'll tell us something 01:21:31.470 --> 01:21:34.830 align:middle line:84% about when a self-adjoint operator is non-negative, 01:21:34.830 --> 01:21:38.820 align:middle line:84% meaning A u inner product u is always non-negative. 01:21:38.820 --> 01:21:40.290 align:middle line:84% And from there, we'll start talking 01:21:40.290 --> 01:21:44.600 align:middle line:84% about the spectral theory for compact self-adjoint operators. 01:21:44.600 --> 01:21:46.350 align:middle line:84% So this is kind of the most complete thing 01:21:46.350 --> 01:21:51.450 align:middle line:84% we can say about the spectrum of a certain class of operators. 01:21:51.450 --> 01:21:55.800 align:middle line:84% And what we can say about self-adjoint compact operators 01:21:55.800 --> 01:22:00.090 align:middle line:84% is that it's pretty close to the finite-dimensional case-- 01:22:00.090 --> 01:22:08.410 align:middle line:84% namely, that the spectrum is that these operators can 01:22:08.410 --> 01:22:09.940 align:middle line:90% essentially be diagonalized. 01:22:09.940 --> 01:22:12.580 align:middle line:84% You can find an orthonormal basis consisting entirely 01:22:12.580 --> 01:22:16.330 align:middle line:84% of eigenvectors for the operator. 01:22:16.330 --> 01:22:18.810 align:middle line:90% And we'll stop there. 01:22:18.810 --> 01:22:24.000 align:middle line:90%