WEBVTT 00:00:00.000 --> 00:00:01.988 align:middle line:90% [SQUEAKING] 00:00:01.988 --> 00:00:03.479 align:middle line:90% [RUSTLING] 00:00:03.479 --> 00:00:05.467 align:middle line:90% [CLICKING] 00:00:05.467 --> 00:00:18.152 align:middle line:90% 00:00:18.152 --> 00:00:19.860 align:middle line:84% PROFESSOR: OK, so we're going to continue 00:00:19.860 --> 00:00:29.365 align:middle line:84% with our discussion of the spectrum of an operator, 00:00:29.365 --> 00:00:31.280 align:middle line:90% a bounded linear operator. 00:00:31.280 --> 00:00:37.270 align:middle line:84% So let me just recall from last time the definition that if A-- 00:00:37.270 --> 00:00:39.370 align:middle line:84% and throughout H is a Hilbert space, 00:00:39.370 --> 00:00:42.490 align:middle line:84% if A is a bounded linear operator, 00:00:42.490 --> 00:00:49.990 align:middle line:84% then the resolvent set of A is the set of all lambdas 00:00:49.990 --> 00:00:52.810 align:middle line:84% and complex numbers such that A minus 00:00:52.810 --> 00:00:56.980 align:middle line:84% lambda times the identity, which I write as A minus lambda, 00:00:56.980 --> 00:01:02.560 align:middle line:84% is bijective, meaning it's 1 to 1 and onto, 00:01:02.560 --> 00:01:04.690 align:middle line:84% which implies by the open mapping theorem, 00:01:04.690 --> 00:01:08.590 align:middle line:84% that it has a bounded inverse or is equivalent to it 00:01:08.590 --> 00:01:10.180 align:middle line:90% having a bounded inverse. 00:01:10.180 --> 00:01:19.070 align:middle line:84% And the spectrum of A was the complement 00:01:19.070 --> 00:01:22.310 align:middle line:84% of the resolvent set within the set of complex numbers. 00:01:22.310 --> 00:01:25.450 align:middle line:90% 00:01:25.450 --> 00:01:30.210 align:middle line:84% And so the spectrum is supposed to be a generalization of, 00:01:30.210 --> 00:01:33.100 align:middle line:84% in finite dimensions, what were called the eigenvalues. 00:01:33.100 --> 00:01:36.940 align:middle line:90% 00:01:36.940 --> 00:01:38.980 align:middle line:90% And so we just-- 00:01:38.980 --> 00:01:46.550 align:middle line:90% 00:01:46.550 --> 00:01:49.700 align:middle line:84% recall we called lambda an element of the spectrum, 00:01:49.700 --> 00:01:58.820 align:middle line:84% an eigenvalue if there exists a u not equal to 0 such 00:01:58.820 --> 00:02:04.160 align:middle line:84% that A minus lambda applied to u is 0. 00:02:04.160 --> 00:02:07.190 align:middle line:84% In other words, A minus lambda is not injective. 00:02:07.190 --> 00:02:11.520 align:middle line:84% So the reason for lambda being in the spectrum is that-- 00:02:11.520 --> 00:02:15.070 align:middle line:84% so for a number to be called an eigenvalue 00:02:15.070 --> 00:02:17.740 align:middle line:84% is that this operator, A minus lambda, 00:02:17.740 --> 00:02:19.250 align:middle line:90% has non-trivial null space. 00:02:19.250 --> 00:02:23.510 align:middle line:84% In other words, it has a non-zero u So 00:02:23.510 --> 00:02:25.220 align:middle line:90% that Au equals lambda u. 00:02:25.220 --> 00:02:33.160 align:middle line:84% And we call this thing an eigenvector. 00:02:33.160 --> 00:02:37.540 align:middle line:84% And we saw last time examples of an operator that 00:02:37.540 --> 00:02:41.170 align:middle line:84% has infinitely many eigenvalues and eigenvectors and also 00:02:41.170 --> 00:02:43.210 align:middle line:84% an example of a bounded linear operator, which 00:02:43.210 --> 00:02:46.330 align:middle line:84% has no eigenvalues and eigenvectors unlike in the case 00:02:46.330 --> 00:02:48.820 align:middle line:84% in finite dimensions, where the spectrum is exactly 00:02:48.820 --> 00:02:51.460 align:middle line:84% the set of eigenvalues of an operator. 00:02:51.460 --> 00:02:55.180 align:middle line:90% 00:02:55.180 --> 00:02:59.150 align:middle line:84% Now, and we also proved at the end of last time-- 00:02:59.150 --> 00:03:02.260 align:middle line:84% so what we could say about these two sets 00:03:02.260 --> 00:03:04.060 align:middle line:84% or what we could say about the spectrum 00:03:04.060 --> 00:03:06.400 align:middle line:90% is that it's a closed set. 00:03:06.400 --> 00:03:11.830 align:middle line:84% And it's contained within the ball of radius norm A 00:03:11.830 --> 00:03:17.470 align:middle line:84% in the complex numbers, which means it's a compact set. 00:03:17.470 --> 00:03:20.110 align:middle line:84% And what we could say, by taking complements 00:03:20.110 --> 00:03:23.200 align:middle line:84% about the resolvent set, is that it's 00:03:23.200 --> 00:03:29.020 align:middle line:84% an open set that contains the exterior to a ball of radius 00:03:29.020 --> 00:03:32.350 align:middle line:84% norm of A in the complex numbers. 00:03:32.350 --> 00:03:37.750 align:middle line:84% And that's about all we can say about the spectrum in general 00:03:37.750 --> 00:03:40.130 align:middle line:90% for now. 00:03:40.130 --> 00:03:43.000 align:middle line:84% But we can say quite a lot about the spectrum 00:03:43.000 --> 00:03:45.980 align:middle line:90% of self-adjoint operators. 00:03:45.980 --> 00:03:49.210 align:middle line:84% And then we can give a pretty complete picture 00:03:49.210 --> 00:03:51.130 align:middle line:84% about the spectrum for compact operators-- 00:03:51.130 --> 00:03:53.680 align:middle line:90% self-adjoint compact operators. 00:03:53.680 --> 00:03:56.830 align:middle line:84% But let's first look at self-adjoint operators. 00:03:56.830 --> 00:04:02.200 align:middle line:84% So at the end of last time, we proved 00:04:02.200 --> 00:04:04.630 align:middle line:84% that if I have a self-adjoint-- and this is not 00:04:04.630 --> 00:04:05.650 align:middle line:90% related to the spectrum. 00:04:05.650 --> 00:04:08.680 align:middle line:84% If I have a self-adjoint bounded linear operator on a Hilbert 00:04:08.680 --> 00:04:18.130 align:middle line:84% space, then for all u, Au, u is a real number. 00:04:18.130 --> 00:04:27.520 align:middle line:84% And we could write the norm of A as this quantity sup 00:04:27.520 --> 00:04:32.490 align:middle line:84% u equals 1 of the absolute value of Au inner product u. 00:04:32.490 --> 00:04:37.410 align:middle line:90% 00:04:37.410 --> 00:04:49.120 align:middle line:84% All right, so now, we have the following theorem 00:04:49.120 --> 00:04:52.390 align:middle line:84% about the spectrum for self-adjoint bounded linear 00:04:52.390 --> 00:04:55.180 align:middle line:90% operators on a Hilbert space. 00:04:55.180 --> 00:04:59.950 align:middle line:90% 00:04:59.950 --> 00:05:06.330 align:middle line:84% So the first is that the spectrum is contained 00:05:06.330 --> 00:05:07.780 align:middle line:90% in the real number line. 00:05:07.780 --> 00:05:17.450 align:middle line:84% So the spectrum of A is contained in the real number 00:05:17.450 --> 00:05:18.680 align:middle line:90% line-- 00:05:18.680 --> 00:05:26.780 align:middle line:84% norm A-- or in this interval, minus norm A to norm A, which 00:05:26.780 --> 00:05:29.870 align:middle line:84% I'm viewing as a subset of the complex numbers, so just 00:05:29.870 --> 00:05:33.245 align:middle line:84% the line segment from minus norm A 00:05:33.245 --> 00:05:35.915 align:middle line:84% to norm A as a subset of the complex numbers. 00:05:35.915 --> 00:05:38.540 align:middle line:90% 00:05:38.540 --> 00:05:42.770 align:middle line:84% And the second is that one of these two endpoints 00:05:42.770 --> 00:05:46.820 align:middle line:84% has to be in the spectrum-- maybe both, but at least one. 00:05:46.820 --> 00:05:52.675 align:middle line:90% 00:05:52.675 --> 00:06:05.840 align:middle line:84% At least one of plus or minus norm A is in the spectrum of A. 00:06:05.840 --> 00:06:16.265 align:middle line:84% OK, so to establish one, we already know-- 00:06:16.265 --> 00:06:20.070 align:middle line:90% 00:06:20.070 --> 00:06:25.160 align:middle line:84% since the spectrum of A is contained 00:06:25.160 --> 00:06:29.360 align:middle line:84% in those complex numbers with modulus less than 00:06:29.360 --> 00:06:33.020 align:middle line:84% or equal to the norm of A, we just 00:06:33.020 --> 00:06:44.610 align:middle line:84% need to show that the spectrum is contained in the real number 00:06:44.610 --> 00:06:47.170 align:middle line:90% line. 00:06:47.170 --> 00:06:50.190 align:middle line:84% Then it must be contained in this interval, 00:06:50.190 --> 00:06:51.525 align:middle line:90% since it's contained here. 00:06:51.525 --> 00:06:58.870 align:middle line:90% 00:06:58.870 --> 00:07:02.950 align:middle line:84% OK, so we'll show that anything off the real number line 00:07:02.950 --> 00:07:04.870 align:middle line:90% lies in the resolvent. 00:07:04.870 --> 00:07:06.565 align:middle line:90% That's how we'll go about this. 00:07:06.565 --> 00:07:09.260 align:middle line:90% 00:07:09.260 --> 00:07:22.200 align:middle line:84% So we'll show that if lambda equals s plus it with t 00:07:22.200 --> 00:07:34.720 align:middle line:84% not equal to 0, then lambda is in the resolvent set of A. 00:07:34.720 --> 00:07:39.500 align:middle line:84% Now, suppose lambda has this form. 00:07:39.500 --> 00:07:47.200 align:middle line:84% Then A minus lambda is equal to A minus s plus-- 00:07:47.200 --> 00:07:56.110 align:middle line:84% or minus, sorry, it, which I can write as A-tilde minus it 00:07:56.110 --> 00:08:00.670 align:middle line:84% with A-tilde, a bounded linear operator given by A minus 00:08:00.670 --> 00:08:07.940 align:middle line:84% s, which is also equal to the adjoint 00:08:07.940 --> 00:08:09.710 align:middle line:90% because s is a real number. 00:08:09.710 --> 00:08:12.200 align:middle line:84% So I should have said t is not equal to 0 00:08:12.200 --> 00:08:15.965 align:middle line:90% and s, t, real numbers. 00:08:15.965 --> 00:08:18.680 align:middle line:90% 00:08:18.680 --> 00:08:25.360 align:middle line:84% So A minus lambda I can write as A-tilde minus it, 00:08:25.360 --> 00:08:29.390 align:middle line:84% where A-tilde is A minus s, again, a self-adjoint operator, 00:08:29.390 --> 00:08:29.950 align:middle line:90% all right? 00:08:29.950 --> 00:08:36.020 align:middle line:84% So if I can do an argument for A-tilde 00:08:36.020 --> 00:08:41.900 align:middle line:84% and show that A-tilde minus it is bijective, 00:08:41.900 --> 00:08:46.790 align:middle line:84% then I can conclude A minus s minus it is bijective. 00:08:46.790 --> 00:08:49.470 align:middle line:90% 00:08:49.470 --> 00:08:51.000 align:middle line:90% Why am I doing this? 00:08:51.000 --> 00:08:56.340 align:middle line:84% Because then I can just focus on the one case that s is 0. 00:08:56.340 --> 00:09:00.460 align:middle line:90% 00:09:00.460 --> 00:09:05.320 align:middle line:84% A minus it is bijective if and only 00:09:05.320 --> 00:09:08.710 align:middle line:90% if A minus lambda is bijective. 00:09:08.710 --> 00:09:12.760 align:middle line:90% 00:09:12.760 --> 00:09:16.330 align:middle line:84% So we only need to consider-- so I can just work on this thing. 00:09:16.330 --> 00:09:19.680 align:middle line:84% But instead of writing A-tilde over and over again, 00:09:19.680 --> 00:09:22.470 align:middle line:84% I'll just switch back to A. So I only 00:09:22.470 --> 00:09:24.680 align:middle line:84% really need to consider the case s equals 0. 00:09:24.680 --> 00:09:37.760 align:middle line:90% 00:09:37.760 --> 00:09:40.940 align:middle line:84% So rather than do the argument for A-tilde minus 00:09:40.940 --> 00:09:43.160 align:middle line:84% it, I'm just going to set s equal to 0 00:09:43.160 --> 00:09:45.370 align:middle line:84% and start doing the argument for A minus it. 00:09:45.370 --> 00:09:55.008 align:middle line:90% 00:09:55.008 --> 00:10:04.380 align:middle line:84% OK, so since by what we proved, or the result from last time-- 00:10:04.380 --> 00:10:13.990 align:middle line:84% so let me just set out what we're going to prove. 00:10:13.990 --> 00:10:21.100 align:middle line:84% If A is self-adjoint, then A minus t minus it 00:10:21.100 --> 00:10:27.970 align:middle line:84% is bijective for all t not equal to 0. 00:10:27.970 --> 00:10:31.550 align:middle line:90% 00:10:31.550 --> 00:10:36.710 align:middle line:84% So once I've proven this claim, then I've proven that-- 00:10:36.710 --> 00:10:40.850 align:middle line:84% I've proven my first part of the theorem. 00:10:40.850 --> 00:10:53.120 align:middle line:84% OK, now, since Au applied to u is real, I get that-- 00:10:53.120 --> 00:10:57.350 align:middle line:84% if I take the imaginary part of A minus 00:10:57.350 --> 00:11:03.230 align:middle line:84% it applied to u, inner product u, this is equal to-- 00:11:03.230 --> 00:11:07.770 align:middle line:90% 00:11:07.770 --> 00:11:10.980 align:middle line:84% Au, u, taking the imaginary part of that is just 0. 00:11:10.980 --> 00:11:27.960 align:middle line:84% So then I get minus t norm u squared, 00:11:27.960 --> 00:11:35.450 align:middle line:84% which implies that since t is non-zero-- 00:11:35.450 --> 00:11:37.160 align:middle line:90% we're assuming t is non-zero-- 00:11:37.160 --> 00:11:41.300 align:middle line:84% A minus it times u equals 0 if and only 00:11:41.300 --> 00:11:46.730 align:middle line:84% if u equals 0 because if this quantity here equals 0, 00:11:46.730 --> 00:11:49.280 align:middle line:90% then this thing here equals 0. 00:11:49.280 --> 00:11:52.250 align:middle line:84% And therefore, the norm of u has to be 0 since t is non-zero. 00:11:52.250 --> 00:11:59.210 align:middle line:84% So the null space of A minus it is just the zero vector. 00:11:59.210 --> 00:12:01.790 align:middle line:84% And therefore, it must-- it's injective, right? 00:12:01.790 --> 00:12:19.957 align:middle line:90% 00:12:19.957 --> 00:12:21.035 align:middle line:90% It is injective. 00:12:21.035 --> 00:12:22.920 align:middle line:90% It is 1 to 1. 00:12:22.920 --> 00:12:25.460 align:middle line:84% All right, now, we just want to show it's bijective-- 00:12:25.460 --> 00:12:28.438 align:middle line:90% or it's surjective. 00:12:28.438 --> 00:12:40.620 align:middle line:84% OK, so similarly, I can prove that the adjoint 00:12:40.620 --> 00:12:51.660 align:middle line:84% of this operator, which is, in fact, A plus it, is injective. 00:12:51.660 --> 00:13:00.910 align:middle line:84% And therefore, I get that the orthogonal complement 00:13:00.910 --> 00:13:04.840 align:middle line:84% to the range of A minus it-- so I want to show this equals 00:13:04.840 --> 00:13:06.670 align:middle line:90% H to show it's surjective. 00:13:06.670 --> 00:13:08.570 align:middle line:84% So the orthogonal complement of that, 00:13:08.570 --> 00:13:14.590 align:middle line:84% which is equal to the null space of the adjoint, which is-- 00:13:14.590 --> 00:13:24.540 align:middle line:90% 00:13:24.540 --> 00:13:32.670 align:middle line:84% equals-- so since this equals 0, I 00:13:32.670 --> 00:13:39.030 align:middle line:84% conclude that the range of A minus it closure, which 00:13:39.030 --> 00:13:45.630 align:middle line:84% is equal to the range of A minus it 00:13:45.630 --> 00:13:48.420 align:middle line:84% taking the orthogonal complement of the orthogonal complement. 00:13:48.420 --> 00:13:49.890 align:middle line:84% So that was part of an assignment 00:13:49.890 --> 00:13:54.660 align:middle line:84% that if I have a subspace of a Hilbert space and I take-- 00:13:54.660 --> 00:13:58.680 align:middle line:84% or let me say it here-- and I take the orthogonal complement 00:13:58.680 --> 00:14:01.530 align:middle line:84% of the orthogonal complement, I don't get back to subspace. 00:14:01.530 --> 00:14:04.290 align:middle line:84% I get the closure of the subspace. 00:14:04.290 --> 00:14:09.480 align:middle line:84% So this is equal to the orthogonal complement 00:14:09.480 --> 00:14:14.250 align:middle line:84% of the zero vector, which is H. So I'll 00:14:14.250 --> 00:14:19.110 align:middle line:84% be done showing that A minus it is surjective if I can show 00:14:19.110 --> 00:14:21.090 align:middle line:84% that the range is closed because then this 00:14:21.090 --> 00:14:34.990 align:middle line:84% will just be the range of A minus it equals H. 00:14:34.990 --> 00:14:43.595 align:middle line:84% So we just need to show now that the range of A minus it 00:14:43.595 --> 00:14:44.095 align:middle line:90% is closed. 00:14:44.095 --> 00:14:50.570 align:middle line:90% 00:14:50.570 --> 00:14:52.970 align:middle line:84% So to show it's closed, we have to show 00:14:52.970 --> 00:14:54.890 align:middle line:84% that if we take a sequence of elements 00:14:54.890 --> 00:14:57.470 align:middle line:84% in here converging to something, then that limit 00:14:57.470 --> 00:14:58.670 align:middle line:90% is, in fact, in the set. 00:14:58.670 --> 00:15:08.460 align:middle line:84% So suppose I have a sequence of elements un such 00:15:08.460 --> 00:15:15.750 align:middle line:84% that A minus it applied to un converges to an element v. So 00:15:15.750 --> 00:15:19.920 align:middle line:84% my goal is to show that v is in the range of A minus it. 00:15:19.920 --> 00:15:23.880 align:middle line:90% 00:15:23.880 --> 00:15:28.035 align:middle line:84% So we want to show v is in the range. 00:15:28.035 --> 00:15:31.983 align:middle line:90% 00:15:31.983 --> 00:15:33.900 align:middle line:84% And then we've shown that the range is closed. 00:15:33.900 --> 00:15:37.155 align:middle line:84% And we're done with the first part. 00:15:37.155 --> 00:15:41.140 align:middle line:90% 00:15:41.140 --> 00:15:45.120 align:middle line:84% So using this argument here, we're 00:15:45.120 --> 00:15:48.780 align:middle line:84% going to show that the un's, which a priori 00:15:48.780 --> 00:15:50.280 align:middle line:84% we don't know converge-- all we know 00:15:50.280 --> 00:15:52.980 align:middle line:84% is that the images of the un's converge. 00:15:52.980 --> 00:15:55.650 align:middle line:84% We are going to show that the un's actually converge. 00:15:55.650 --> 00:15:57.780 align:middle line:84% And then that will essentially finish the proof. 00:15:57.780 --> 00:16:01.670 align:middle line:90% 00:16:01.670 --> 00:16:08.280 align:middle line:84% Then we have that the absolute value of t u minus um 00:16:08.280 --> 00:16:09.830 align:middle line:90% norm squared-- 00:16:09.830 --> 00:16:12.800 align:middle line:84% this is, by this calculation we did over here, 00:16:12.800 --> 00:16:16.490 align:middle line:84% equal to the absolute value of the image of-- 00:16:16.490 --> 00:16:28.070 align:middle line:84% I mean, the imaginary part of A minus it un minus um un 00:16:28.070 --> 00:16:28.951 align:middle line:90% minus um. 00:16:28.951 --> 00:16:31.897 align:middle line:90% 00:16:31.897 --> 00:16:35.790 align:middle line:90% Take the value of all of that. 00:16:35.790 --> 00:16:38.870 align:middle line:84% And now, this is less than or equal 00:16:38.870 --> 00:16:41.870 align:middle line:84% to-- so the absolute value of the imaginary part 00:16:41.870 --> 00:16:45.830 align:middle line:84% of a complex number is less than or equal to the absolute value 00:16:45.830 --> 00:16:48.530 align:middle line:84% of that complex number, which by Cauchy-Schwarz I 00:16:48.530 --> 00:16:52.610 align:middle line:84% can say is less than or equal to A minus it 00:16:52.610 --> 00:16:55.400 align:middle line:90% applied to un minus um. 00:16:55.400 --> 00:17:07.025 align:middle line:84% But I'll write it as A minus it times the norm of un minus um. 00:17:07.025 --> 00:17:11.960 align:middle line:84% And I started off with t, which is non-zero, 00:17:11.960 --> 00:17:14.640 align:middle line:84% times the norm of un minus um squared. 00:17:14.640 --> 00:17:16.939 align:middle line:90% So I get that un minus um-- 00:17:16.939 --> 00:17:20.869 align:middle line:90% 00:17:20.869 --> 00:17:22.880 align:middle line:84% that this is less than or equal to 1 00:17:22.880 --> 00:17:28.250 align:middle line:84% over the absolute value of t times the norm 00:17:28.250 --> 00:17:35.600 align:middle line:84% A minus it applied to un minus A minus it applied to um norm. 00:17:35.600 --> 00:17:40.830 align:middle line:90% 00:17:40.830 --> 00:17:43.500 align:middle line:90% Now, this thing on the right-- 00:17:43.500 --> 00:17:46.260 align:middle line:90% 00:17:46.260 --> 00:17:49.590 align:middle line:84% or I should say A minus it applied to un, 00:17:49.590 --> 00:17:52.110 align:middle line:90% this is a convergent sequence. 00:17:52.110 --> 00:17:54.480 align:middle line:84% In particular, it's a Cauchy sequence. 00:17:54.480 --> 00:17:58.170 align:middle line:84% So given epsilon, I can find capital N 00:17:58.170 --> 00:18:01.470 align:middle line:84% so that the norm of this right-hand side 00:18:01.470 --> 00:18:05.940 align:middle line:84% is less than epsilon times the magnitude of t. 00:18:05.940 --> 00:18:09.060 align:middle line:84% And therefore, for all capital N bigger than or equal to-- 00:18:09.060 --> 00:18:12.240 align:middle line:84% or for all little n m bigger than or equal to that capital 00:18:12.240 --> 00:18:15.630 align:middle line:84% N, this in norm will be less than epsilon. 00:18:15.630 --> 00:18:24.660 align:middle line:90% 00:18:24.660 --> 00:18:34.970 align:middle line:84% Since this is Cauchy because it's convergent, 00:18:34.970 --> 00:18:43.230 align:middle line:84% the previous estimate implies that the sequence un is Cauchy. 00:18:43.230 --> 00:18:45.600 align:middle line:84% And since we're in a Hilbert space, which 00:18:45.600 --> 00:18:50.170 align:middle line:84% means it's complete, we can find a limit of this un. 00:18:50.170 --> 00:18:57.360 align:middle line:84% There exists a u in H such that un converges to u. 00:18:57.360 --> 00:18:58.860 align:middle line:90% And then we're done now. 00:18:58.860 --> 00:19:07.050 align:middle line:84% Then since A is a bounded linear operator, A minus it u-- 00:19:07.050 --> 00:19:10.110 align:middle line:84% or A minus it applied to u, this is equal to the limit 00:19:10.110 --> 00:19:16.230 align:middle line:84% as n goes to infinity of A minus it applied to un. 00:19:16.230 --> 00:19:20.310 align:middle line:84% But remember, we assumed that this converges to some element 00:19:20.310 --> 00:19:27.270 align:middle line:84% v. And therefore, v is equal to something in the image 00:19:27.270 --> 00:19:28.995 align:middle line:90% or in the range of A minus it. 00:19:28.995 --> 00:19:31.840 align:middle line:90% 00:19:31.840 --> 00:19:32.815 align:middle line:90% v is in the range. 00:19:32.815 --> 00:19:38.760 align:middle line:90% 00:19:38.760 --> 00:19:41.980 align:middle line:84% And thus, the range of A minus it is closed. 00:19:41.980 --> 00:19:48.630 align:middle line:84% And by this here, we conclude that the range of A minus t 00:19:48.630 --> 00:19:51.870 align:middle line:84% equals H. So A minus it is bijective. 00:19:51.870 --> 00:20:00.590 align:middle line:90% 00:20:00.590 --> 00:20:03.470 align:middle line:84% And that concludes the proof of the first property 00:20:03.470 --> 00:20:04.490 align:middle line:90% we wanted to do. 00:20:04.490 --> 00:20:13.440 align:middle line:90% 00:20:13.440 --> 00:20:19.680 align:middle line:84% OK, so for the second thing we wanted 00:20:19.680 --> 00:20:25.470 align:middle line:84% to show, we wanted to show that at least one 00:20:25.470 --> 00:20:37.050 align:middle line:84% of plus or minus norm of A is in the spectrum of A. Now, 00:20:37.050 --> 00:20:46.920 align:middle line:84% since the norm of A is equal to the sup over norm of u 00:20:46.920 --> 00:20:53.000 align:middle line:84% equals 1 of the absolute value of Au, u-- 00:20:53.000 --> 00:20:55.040 align:middle line:84% so a supremum is always characterized 00:20:55.040 --> 00:21:00.530 align:middle line:84% by being an upper bound and also there existing a sequence 00:21:00.530 --> 00:21:02.630 align:middle line:84% in the set of things you're taking the supremum 00:21:02.630 --> 00:21:07.400 align:middle line:90% of converging to that supremum. 00:21:07.400 --> 00:21:12.620 align:middle line:84% So that implies that there exists a sequence of unit 00:21:12.620 --> 00:21:22.650 align:middle line:84% vectors so that Au inner product u in absolute value 00:21:22.650 --> 00:21:24.810 align:middle line:84% has to converge to the norm of a. 00:21:24.810 --> 00:21:29.610 align:middle line:84% So in particular, Au applied to u 00:21:29.610 --> 00:21:41.132 align:middle line:84% has to converge to the norm of A or minus norm of A 00:21:41.132 --> 00:21:43.133 align:middle line:90% as n goes to infinity. 00:21:43.133 --> 00:21:44.050 align:middle line:90% And there's no n here. 00:21:44.050 --> 00:22:01.880 align:middle line:90% 00:22:01.880 --> 00:22:09.790 align:middle line:84% All right, then what does this imply? 00:22:09.790 --> 00:22:16.870 align:middle line:84% Then this implies that, for at least one of these choices, 00:22:16.870 --> 00:22:35.540 align:middle line:84% then A plus or minus norm of A applied to un inner product un 00:22:35.540 --> 00:22:38.970 align:middle line:84% converges to 0 as n goes to infinity, 00:22:38.970 --> 00:22:42.590 align:middle line:84% where here the plus or minus is chosen depending 00:22:42.590 --> 00:22:46.430 align:middle line:84% on whether this goes to plus or minus the norm of A. 00:22:46.430 --> 00:22:50.300 align:middle line:84% So this sign here would be the opposite of whichever sign 00:22:50.300 --> 00:22:52.010 align:middle line:90% this sequence goes to. 00:22:52.010 --> 00:23:00.540 align:middle line:90% 00:23:00.540 --> 00:23:07.020 align:middle line:84% I now claim that this property here implies that whichever 00:23:07.020 --> 00:23:13.200 align:middle line:84% sign we have that for, that this operator appearing here cannot 00:23:13.200 --> 00:23:13.890 align:middle line:90% be invertible. 00:23:13.890 --> 00:23:19.300 align:middle line:84% And therefore, whichever sign appeared here or the opposite 00:23:19.300 --> 00:23:23.020 align:middle line:84% sign-- so let me, in fact, stay-- 00:23:23.020 --> 00:23:25.340 align:middle line:84% so the minus corresponds to the plus sign. 00:23:25.340 --> 00:23:28.110 align:middle line:84% The plus sign corresponds to the minus sign 00:23:28.110 --> 00:23:30.130 align:middle line:90% if we have one of those. 00:23:30.130 --> 00:23:33.540 align:middle line:84% So I claim that this property here 00:23:33.540 --> 00:23:37.800 align:middle line:84% implies that this operator is not invertible. 00:23:37.800 --> 00:23:43.150 align:middle line:84% And therefore, one of those is in the spectrum. 00:23:43.150 --> 00:23:51.510 align:middle line:84% So suppose instead that A minus plus norm of a, whichever 00:23:51.510 --> 00:23:53.520 align:middle line:90% one appeared, is invertible. 00:23:53.520 --> 00:23:57.110 align:middle line:90% 00:23:57.110 --> 00:24:01.610 align:middle line:84% So whichever one does satisfy this is invertible. 00:24:01.610 --> 00:24:04.400 align:middle line:90% 00:24:04.400 --> 00:24:08.930 align:middle line:90% Then the un's all have norm 1. 00:24:08.930 --> 00:24:12.860 align:middle line:90% So 1 is equal to the norm of un. 00:24:12.860 --> 00:24:19.130 align:middle line:84% And I can write this as the norm of A minus plus norm 00:24:19.130 --> 00:24:25.490 align:middle line:84% A inverse applied to A minus plus norm A, 00:24:25.490 --> 00:24:30.155 align:middle line:84% because that's just the identity, applied to un. 00:24:30.155 --> 00:24:32.420 align:middle line:84% And this is less than or equal to the norm 00:24:32.420 --> 00:24:42.810 align:middle line:84% of the inverse times the norm of this quantity. 00:24:42.810 --> 00:24:51.500 align:middle line:90% 00:24:51.500 --> 00:24:54.520 align:middle line:90% And so this is a fixed number. 00:24:54.520 --> 00:24:56.440 align:middle line:84% And this thing is converging to 0. 00:24:56.440 --> 00:24:59.140 align:middle line:84% So the right-hand side converges to zero. 00:24:59.140 --> 00:25:01.900 align:middle line:90% But that's 1, right? 00:25:01.900 --> 00:25:03.550 align:middle line:84% I have 1 is less than or equal to 0. 00:25:03.550 --> 00:25:06.310 align:middle line:90% So that's a contradiction. 00:25:06.310 --> 00:25:13.990 align:middle line:84% Thus, A minus or plus the norm of A-- again, 00:25:13.990 --> 00:25:18.070 align:middle line:84% the minus or plus corresponds to which sign of the norm of A we 00:25:18.070 --> 00:25:19.585 align:middle line:84% had that sequence converging to-- 00:25:19.585 --> 00:25:24.230 align:middle line:90% 00:25:24.230 --> 00:25:33.430 align:middle line:84% is not invertible, which implies that plus or minus at least one 00:25:33.430 --> 00:25:47.420 align:middle line:84% of these is in the spectrum of A. 00:25:47.420 --> 00:25:52.570 align:middle line:84% OK, now, we can, in fact, do a little bit better then-- 00:25:52.570 --> 00:25:56.582 align:middle line:90% 00:25:56.582 --> 00:25:58.790 align:middle line:84% based on this argument, we can do a little bit better 00:25:58.790 --> 00:26:03.170 align:middle line:84% in bounding the spectrum of a self-adjoint operator 00:26:03.170 --> 00:26:09.190 align:middle line:84% than just the bound that we have coming from the general theory. 00:26:09.190 --> 00:26:13.130 align:middle line:90% 00:26:13.130 --> 00:26:16.240 align:middle line:90% So what do I mean by that? 00:26:16.240 --> 00:26:24.340 align:middle line:90% 00:26:24.340 --> 00:26:38.830 align:middle line:84% So if A is a self-adjoint bounded operator 00:26:38.830 --> 00:26:45.760 align:middle line:84% and a-minus is equal to the infimum over all u equals 00:26:45.760 --> 00:27:01.330 align:middle line:84% 1 Au applied to u, a-plus equals the sup Au applied to u, 00:27:01.330 --> 00:27:09.140 align:middle line:84% then the spectrum of A is contained. 00:27:09.140 --> 00:27:13.075 align:middle line:84% So first off then, a couple of things, two things-- 00:27:13.075 --> 00:27:18.050 align:middle line:90% 00:27:18.050 --> 00:27:24.530 align:middle line:84% then both of these numbers are in the spectrum of A. 00:27:24.530 --> 00:27:28.905 align:middle line:84% And the spectrum is contained in this line segment. 00:27:28.905 --> 00:27:36.030 align:middle line:90% 00:27:36.030 --> 00:27:41.160 align:middle line:84% So this is something of a tighter bound because a-minus 00:27:41.160 --> 00:27:44.400 align:middle line:84% is always bigger than or equal to minus the norm of A just 00:27:44.400 --> 00:27:48.015 align:middle line:84% by this always being bounded below by the norm of A. 00:27:48.015 --> 00:27:51.180 align:middle line:84% And a-plus is always bounded above by the norm of A, 00:27:51.180 --> 00:27:53.400 align:middle line:84% since this is always bounded above by the norm of A. 00:27:53.400 --> 00:27:56.950 align:middle line:84% So the sup will be bounded above by that. 00:27:56.950 --> 00:28:01.200 align:middle line:84% So this is a tighter estimate than just the regular estimate 00:28:01.200 --> 00:28:04.320 align:middle line:84% that says the spectrum is contained inside 00:28:04.320 --> 00:28:09.290 align:middle line:84% of the interval from minus norm A to norm A. 00:28:09.290 --> 00:28:12.810 align:middle line:84% And in fact, you get more information, that not just one 00:28:12.810 --> 00:28:17.520 align:middle line:84% of the endpoints have to be in, but both of these endpoints 00:28:17.520 --> 00:28:18.420 align:middle line:90% are in the spectrum. 00:28:18.420 --> 00:28:25.050 align:middle line:90% 00:28:25.050 --> 00:28:30.840 align:middle line:84% So the proof of this is just kind 00:28:30.840 --> 00:28:36.930 align:middle line:84% of a trick of using what we've done already. 00:28:36.930 --> 00:28:44.540 align:middle line:90% So first, note that-- 00:28:44.540 --> 00:28:50.010 align:middle line:84% again, since the absolute value of Au inner product 00:28:50.010 --> 00:29:02.140 align:middle line:84% is always less than or equal to the norm of A, 00:29:02.140 --> 00:29:07.540 align:middle line:84% for all unit vectors, this implies that this quantity here 00:29:07.540 --> 00:29:12.500 align:middle line:84% is always bounded below by norm of A 00:29:12.500 --> 00:29:19.150 align:middle line:84% and bounded above by norm of A. And therefore, the infimum 00:29:19.150 --> 00:29:22.990 align:middle line:84% of this is always bounded from-- so this is a lower bound 00:29:22.990 --> 00:29:25.010 align:middle line:90% for this quantity here. 00:29:25.010 --> 00:29:26.920 align:middle line:84% So this infimum is bigger than it, 00:29:26.920 --> 00:29:30.460 align:middle line:84% since it's the greatest lower bound. 00:29:30.460 --> 00:29:34.540 align:middle line:84% And the least upper bound of these quantities 00:29:34.540 --> 00:29:36.910 align:middle line:84% is always less than or equal to the norm of A. 00:29:36.910 --> 00:29:44.280 align:middle line:84% So these are actual numbers for one. 00:29:44.280 --> 00:29:57.020 align:middle line:84% OK, now, by the definition of a-plus or minus, 00:29:57.020 --> 00:30:04.940 align:middle line:84% there exists sequences of unit vectors un-plus or minus such 00:30:04.940 --> 00:30:08.720 align:middle line:84% that A applied to un-plus or minus inner product 00:30:08.720 --> 00:30:15.140 align:middle line:84% un-plus or minus converges to a-plus or minus. 00:30:15.140 --> 00:30:21.380 align:middle line:84% Now, by the argument we just gave with a-plus or minus being 00:30:21.380 --> 00:30:23.750 align:middle line:90% the norm of A-- 00:30:23.750 --> 00:30:32.480 align:middle line:84% but now, we have this property, i.e. 00:30:32.480 --> 00:30:38.420 align:middle line:84% A minus a-plus or minus applied to un-plus or minus, un-plus 00:30:38.420 --> 00:30:42.650 align:middle line:90% or minus converges to 0. 00:30:42.650 --> 00:30:46.370 align:middle line:84% Since I have this property by the previous argument I gave, 00:30:46.370 --> 00:30:49.700 align:middle line:84% this implies that both a-plus and a-minus 00:30:49.700 --> 00:30:53.770 align:middle line:84% are in the spectrum of A, since we 00:30:53.770 --> 00:30:57.760 align:middle line:84% have for each choice of plus or minus a sequence of unit 00:30:57.760 --> 00:31:00.910 align:middle line:84% vectors so that this quantity here goes to 0. 00:31:00.910 --> 00:31:03.160 align:middle line:84% A minute ago, we could just assert that there 00:31:03.160 --> 00:31:05.240 align:middle line:90% was a sequence of unit vectors. 00:31:05.240 --> 00:31:09.070 align:middle line:84% So for at least one choice of plus or minus the norm of A, 00:31:09.070 --> 00:31:11.050 align:middle line:90% we had this thing going to 0. 00:31:11.050 --> 00:31:15.700 align:middle line:84% But for these two numbers, because it's the inf 00:31:15.700 --> 00:31:21.000 align:middle line:84% and because this is the sup, we can always find unit vectors 00:31:21.000 --> 00:31:23.450 align:middle line:84% so that this quantity is converging to the sup, which 00:31:23.450 --> 00:31:25.152 align:middle line:84% is a-plus; this quantity is converging 00:31:25.152 --> 00:31:26.360 align:middle line:90% to the inf, which is a-minus. 00:31:26.360 --> 00:31:29.630 align:middle line:90% 00:31:29.630 --> 00:31:37.640 align:middle line:84% So by the previous argument, we get 00:31:37.640 --> 00:31:45.538 align:middle line:84% that both a-plus or minus are in the spectrum of A. Again, 00:31:45.538 --> 00:31:46.580 align:middle line:90% I just want to emphasize. 00:31:46.580 --> 00:31:51.710 align:middle line:84% Before, we could just say that one of the norms of A or plus 00:31:51.710 --> 00:31:52.580 align:middle line:90% or-- 00:31:52.580 --> 00:31:55.790 align:middle line:84% at least one choice of plus or minus the norm of A 00:31:55.790 --> 00:31:56.780 align:middle line:90% is in the spectrum. 00:31:56.780 --> 00:32:00.120 align:middle line:84% Here, we're saying both of these numbers are in the spectrum. 00:32:00.120 --> 00:32:02.640 align:middle line:84% So now, what remains is to show that the spectrum is, 00:32:02.640 --> 00:32:05.662 align:middle line:84% in fact, contained in this interval 00:32:05.662 --> 00:32:06.620 align:middle line:90% from a-minus to a-plus. 00:32:06.620 --> 00:32:13.790 align:middle line:90% 00:32:13.790 --> 00:32:20.200 align:middle line:84% All right, so let b be their midpoint. 00:32:20.200 --> 00:32:26.310 align:middle line:90% 00:32:26.310 --> 00:32:36.120 align:middle line:84% And B equals A minus b times I. Now, 00:32:36.120 --> 00:32:39.640 align:middle line:84% B's a real number because those are two real numbers. 00:32:39.640 --> 00:32:42.360 align:middle line:84% So capital B is the difference between A 00:32:42.360 --> 00:32:46.450 align:middle line:84% and-- is A minus a real number times the identity. 00:32:46.450 --> 00:32:58.600 align:middle line:84% So B is self-adjoint and a bounded linear operator on H. 00:32:58.600 --> 00:33:17.880 align:middle line:84% So by the previous theorem, we get that the spectrum of B, 00:33:17.880 --> 00:33:25.445 align:middle line:84% well, is contained in the norm of B-- 00:33:25.445 --> 00:33:29.730 align:middle line:90% 00:33:29.730 --> 00:33:39.140 align:middle line:84% so minus norm of B, norm of B. And it 00:33:39.140 --> 00:33:43.670 align:middle line:84% shouldn't take much thought to realize 00:33:43.670 --> 00:33:48.260 align:middle line:84% that if the spectrum of B, which is a shift of a by little b, 00:33:48.260 --> 00:33:54.060 align:middle line:84% is contained in this interval, then the spec of A 00:33:54.060 --> 00:33:58.020 align:middle line:84% is contained in minus norm of B plus little 00:33:58.020 --> 00:34:01.800 align:middle line:90% b, normal of B plus b. 00:34:01.800 --> 00:34:04.770 align:middle line:90% 00:34:04.770 --> 00:34:09.270 align:middle line:84% So now, what's left is to compute the norm B, all right? 00:34:09.270 --> 00:34:17.610 align:middle line:84% But this is not too difficult. We have that the norm of B-- 00:34:17.610 --> 00:34:26.250 align:middle line:84% this is equal to the sup of u equals 1 Bu applied to u. 00:34:26.250 --> 00:34:34.560 align:middle line:84% And now, I take the sup over all u equals 1. 00:34:34.560 --> 00:34:37.050 align:middle line:84% And let me plug in what A is and B is. 00:34:37.050 --> 00:34:42.840 align:middle line:84% And this is Au, u minus a-plus minus-- 00:34:42.840 --> 00:34:45.420 align:middle line:90% or a-plus minus a-minus over 2. 00:34:45.420 --> 00:34:49.690 align:middle line:90% 00:34:49.690 --> 00:34:52.060 align:middle line:90% Now, here's the picture. 00:34:52.060 --> 00:34:53.290 align:middle line:90% Here's a-minus. 00:34:53.290 --> 00:34:55.230 align:middle line:90% Here's a-plus. 00:34:55.230 --> 00:34:58.780 align:middle line:84% a-plus is the sup over all of these expressions 00:34:58.780 --> 00:35:01.150 align:middle line:90% where u has unit length. 00:35:01.150 --> 00:35:03.850 align:middle line:84% a-minus is the inf over all these expressions, 00:35:03.850 --> 00:35:05.320 align:middle line:90% where u has unit length. 00:35:05.320 --> 00:35:09.620 align:middle line:84% a-plus plus a-minus is the point right in the middle of them. 00:35:09.620 --> 00:35:12.880 align:middle line:84% So what's the biggest this-- or what's 00:35:12.880 --> 00:35:18.840 align:middle line:84% the supremum of the difference between these numbers 00:35:18.840 --> 00:35:19.930 align:middle line:90% and the midpoint? 00:35:19.930 --> 00:35:26.030 align:middle line:84% Well, it's the distance given by the distance from a-plus 00:35:26.030 --> 00:35:28.530 align:middle line:84% to the midpoint, which is equal to the distance from a-minus 00:35:28.530 --> 00:35:37.080 align:middle line:84% to the midpoint, which is a-plus minus a-minus over 2. 00:35:37.080 --> 00:35:43.180 align:middle line:90% 00:35:43.180 --> 00:35:45.820 align:middle line:84% And since that's the norm of B, when we plug that 00:35:45.820 --> 00:35:51.330 align:middle line:84% into what we had a minute ago, we 00:35:51.330 --> 00:35:56.940 align:middle line:84% conclude that the spectrum is contained in a-minus, a-plus. 00:35:56.940 --> 00:36:12.240 align:middle line:90% 00:36:12.240 --> 00:36:16.213 align:middle line:84% So as a simple corollary of what we've done, 00:36:16.213 --> 00:36:17.630 align:middle line:84% we have this nice little statement 00:36:17.630 --> 00:36:22.220 align:middle line:84% about when exactly a self-adjoint bounded linear 00:36:22.220 --> 00:36:24.560 align:middle line:90% operator is non-negative. 00:36:24.560 --> 00:36:34.080 align:middle line:84% So let A be a self-adjoint bounded linear operator 00:36:34.080 --> 00:36:36.360 align:middle line:90% on a Hilbert space. 00:36:36.360 --> 00:36:46.440 align:middle line:84% Then for all u, Au inner product u 00:36:46.440 --> 00:36:53.380 align:middle line:84% is bigger than or equal to 0 if and only if the spectrum of A 00:36:53.380 --> 00:37:00.310 align:middle line:84% is contained in the non-negative numbers. 00:37:00.310 --> 00:37:08.750 align:middle line:90% 00:37:08.750 --> 00:37:11.692 align:middle line:84% So I'm not even going to write out the proof. 00:37:11.692 --> 00:37:13.400 align:middle line:84% I'm just going to talk my way through it. 00:37:13.400 --> 00:37:18.540 align:middle line:90% 00:37:18.540 --> 00:37:23.970 align:middle line:84% So let's suppose that Au inner product u is non-negative. 00:37:23.970 --> 00:37:28.520 align:middle line:84% Then this number a-minus is non-negative. 00:37:28.520 --> 00:37:32.300 align:middle line:84% And therefore, the spectrum is contained in a-minus, a-plus, 00:37:32.300 --> 00:37:34.805 align:middle line:84% which is the subset of the non-negative real numbers. 00:37:34.805 --> 00:37:37.530 align:middle line:90% 00:37:37.530 --> 00:37:39.710 align:middle line:84% On the other hand, suppose that the spectrum of A 00:37:39.710 --> 00:37:42.540 align:middle line:90% is contained in here. 00:37:42.540 --> 00:37:46.530 align:middle line:84% Then a-minus, which is in the spectrum, 00:37:46.530 --> 00:37:50.040 align:middle line:84% has to be in the set of non-negative real numbers. 00:37:50.040 --> 00:37:52.110 align:middle line:84% And therefore, Au inner product u always 00:37:52.110 --> 00:37:54.060 align:middle line:84% has to be non-negative, since a-minus 00:37:54.060 --> 00:37:55.335 align:middle line:90% is the inf over all of these. 00:37:55.335 --> 00:37:58.470 align:middle line:90% 00:37:58.470 --> 00:38:04.890 align:middle line:84% So now, we're going to move on to the spectral theory for not 00:38:04.890 --> 00:38:07.380 align:middle line:84% just self-adjoint operators, but self-adjoint operators 00:38:07.380 --> 00:38:08.370 align:middle line:90% that are also compact. 00:38:08.370 --> 00:38:15.720 align:middle line:90% 00:38:15.720 --> 00:38:18.960 align:middle line:84% Again, a natural example is given 00:38:18.960 --> 00:38:24.210 align:middle line:84% by the inverse of taking the second derivative along 00:38:24.210 --> 00:38:27.450 align:middle line:84% with requiring 0 at the endpoints, this operator I 00:38:27.450 --> 00:38:29.700 align:middle line:90% gave last time. 00:38:29.700 --> 00:38:32.205 align:middle line:90% That is a bounded self-adjoint-- 00:38:32.205 --> 00:38:35.950 align:middle line:84% or a compact self-adjoint operator. 00:38:35.950 --> 00:38:40.600 align:middle line:84% So all the spectral theory we developed for that applies. 00:38:40.600 --> 00:38:43.600 align:middle line:84% And the spectrum for that operator ends up being 1 over 00:38:43.600 --> 00:38:47.050 align:middle line:84% the eigenvalues corresponding to u-double prime equals lambda-- 00:38:47.050 --> 00:38:50.830 align:middle line:84% equals, say, mu times u with 0 at the endpoint. 00:38:50.830 --> 00:38:53.950 align:middle line:90% 00:38:53.950 --> 00:38:55.600 align:middle line:84% And you'll see that in the assignment. 00:38:55.600 --> 00:38:57.500 align:middle line:84% Or maybe I'll do it as an example. 00:38:57.500 --> 00:39:12.940 align:middle line:84% So now, we're moving on to spectral theory for compact 00:39:12.940 --> 00:39:17.500 align:middle line:84% self-adjoint operators, which is one of the most, again, 00:39:17.500 --> 00:39:18.580 align:middle line:90% complete things-- 00:39:18.580 --> 00:39:24.130 align:middle line:84% or class of operators we can say the most about when 00:39:24.130 --> 00:39:25.930 align:middle line:90% it comes to the spectrum. 00:39:25.930 --> 00:39:31.340 align:middle line:84% And I'll go ahead and give you a preview 00:39:31.340 --> 00:39:36.650 align:middle line:84% of what we can say about the spectrum for these operators, 00:39:36.650 --> 00:39:38.840 align:middle line:84% that it essentially consists of nothing 00:39:38.840 --> 00:39:48.350 align:middle line:84% but eigenvalues with the possible exception of 0 00:39:48.350 --> 00:39:51.870 align:middle line:84% being an accumulation point of the eigenvalues. 00:39:51.870 --> 00:39:54.410 align:middle line:84% So what we'll prove is that the spectrum 00:39:54.410 --> 00:39:56.510 align:middle line:84% of a compact self-adjoint operator 00:39:56.510 --> 00:40:01.920 align:middle line:84% consists of the eigenvalues of this operator 00:40:01.920 --> 00:40:04.485 align:middle line:90% along with possibly 0. 00:40:04.485 --> 00:40:10.290 align:middle line:90% 00:40:10.290 --> 00:40:16.510 align:middle line:84% And 0 may or may not be an eigenvalue. 00:40:16.510 --> 00:40:18.760 align:middle line:84% If it's not an eigenvalue, then it's 00:40:18.760 --> 00:40:21.190 align:middle line:90% the limit of the eigenvalues. 00:40:21.190 --> 00:40:26.020 align:middle line:84% And in fact, implicit in that statement 00:40:26.020 --> 00:40:29.380 align:middle line:84% is that the spectrum is, in fact, countable 00:40:29.380 --> 00:40:31.940 align:middle line:84% for a compact self-adjoint operator. 00:40:31.940 --> 00:40:36.900 align:middle line:90% So why should we expect that? 00:40:36.900 --> 00:40:39.708 align:middle line:84% Or why should we expect such a complete picture? 00:40:39.708 --> 00:40:41.250 align:middle line:84% In the end, we'll also prove that you 00:40:41.250 --> 00:40:44.010 align:middle line:84% can find a basis for H consisting entirely 00:40:44.010 --> 00:40:50.320 align:middle line:84% of eigenvectors of the operator A, which is, again, 00:40:50.320 --> 00:40:56.870 align:middle line:84% a generalization to infinite dimensions of what hopefully 00:40:56.870 --> 00:40:58.215 align:middle line:90% you saw in finite dimensions. 00:40:58.215 --> 00:40:59.840 align:middle line:84% But if you didn't, our proof will still 00:40:59.840 --> 00:41:01.220 align:middle line:90% apply to finite dimensions. 00:41:01.220 --> 00:41:05.840 align:middle line:84% So why should that then apply to compact self-adjoint operators 00:41:05.840 --> 00:41:07.520 align:middle line:84% if you believe it for finite dimensions? 00:41:07.520 --> 00:41:10.970 align:middle line:84% Well, it's because, again, compact self-adjoint operators 00:41:10.970 --> 00:41:15.047 align:middle line:84% are the norm limit of finite rank operators, all right? 00:41:15.047 --> 00:41:16.880 align:middle line:84% And finite rank operators, again, these just 00:41:16.880 --> 00:41:18.530 align:middle line:84% correspond to basically matrices. 00:41:18.530 --> 00:41:22.850 align:middle line:84% We know how to compute the eigenvalues of matrices. 00:41:22.850 --> 00:41:24.920 align:middle line:84% For finite rank operators, they could 00:41:24.920 --> 00:41:33.200 align:middle line:84% have a very large null space, meaning the eigenvalue 0 could 00:41:33.200 --> 00:41:35.330 align:middle line:90% have a very large eigenspace. 00:41:35.330 --> 00:41:39.560 align:middle line:84% But that's the point of why you expect maybe 00:41:39.560 --> 00:41:41.120 align:middle line:84% things to carry over to the setting 00:41:41.120 --> 00:41:42.890 align:middle line:84% of compact self-adjoint operators 00:41:42.890 --> 00:41:47.430 align:middle line:84% from what you know in finite dimensions. 00:41:47.430 --> 00:41:58.765 align:middle line:84% OK, so this is not so much a definition as just notation. 00:41:58.765 --> 00:42:04.200 align:middle line:90% 00:42:04.200 --> 00:42:09.320 align:middle line:84% If A is a bounded linear operator, 00:42:09.320 --> 00:42:18.630 align:middle line:84% I will denote E lambda to be the null space of A minus lambda-- 00:42:18.630 --> 00:42:20.940 align:middle line:90% in other words, the set of-- 00:42:20.940 --> 00:42:26.280 align:middle line:84% or the subspace of eigenvectors with eigenvalue lambda, which, 00:42:26.280 --> 00:42:34.820 align:middle line:84% again, is the set of u in H such that A minus lambda u equals 0. 00:42:34.820 --> 00:42:41.130 align:middle line:90% 00:42:41.130 --> 00:42:52.710 align:middle line:84% So first off, before we get to classifying the spectral-- 00:42:52.710 --> 00:42:55.140 align:middle line:84% or the spectrum of a compact self-adjoint operator 00:42:55.140 --> 00:42:59.400 align:middle line:84% as basically consisting of eigenvalues along with 0, 00:42:59.400 --> 00:43:04.530 align:middle line:84% we'll first give some kind of general properties 00:43:04.530 --> 00:43:10.930 align:middle line:84% of eigenvalues in general for a compact self-adjoint operator. 00:43:10.930 --> 00:43:23.610 align:middle line:84% So we have the following theorem that suppose A-star in A 00:43:23.610 --> 00:43:25.740 align:middle line:84% is a compact self-adjoint operator. 00:43:25.740 --> 00:43:35.120 align:middle line:90% 00:43:35.120 --> 00:43:38.195 align:middle line:90% Then a few things-- 00:43:38.195 --> 00:43:42.060 align:middle line:90% 00:43:42.060 --> 00:43:51.430 align:middle line:84% if lambda not equal to 0 is an eigenvalue of A, 00:43:51.430 --> 00:43:57.330 align:middle line:84% then the dimension of E lambda, the eigenspace, 00:43:57.330 --> 00:43:59.890 align:middle line:84% the linear subspace of all vectors 00:43:59.890 --> 00:44:06.910 align:middle line:84% that are eigenvectors of A, this is finite. 00:44:06.910 --> 00:44:10.625 align:middle line:84% So for a given eigenvalue, the dimension of the eigenspace 00:44:10.625 --> 00:44:11.125 align:middle line:90% is finite. 00:44:11.125 --> 00:44:15.770 align:middle line:90% 00:44:15.770 --> 00:44:21.290 align:middle line:84% The second is that if I take two different eigenvalues, 00:44:21.290 --> 00:44:24.650 align:middle line:84% the corresponding eigenspaces are 00:44:24.650 --> 00:44:26.150 align:middle line:90% perpendicular to each other. 00:44:26.150 --> 00:44:30.510 align:middle line:90% 00:44:30.510 --> 00:44:32.230 align:middle line:84% Lambda 1 does not equal lambda 2. 00:44:32.230 --> 00:44:36.740 align:middle line:90% 00:44:36.740 --> 00:44:46.130 align:middle line:84% For eigenvalues of A, then E lambda 1, 00:44:46.130 --> 00:44:56.930 align:middle line:84% E lambda 2 are orthogonal or perpendicular. 00:44:56.930 --> 00:45:01.040 align:middle line:84% Every element in E lambda 1 is orthogonal to every element 00:45:01.040 --> 00:45:02.810 align:middle line:90% in E lambda 2 and vice versa. 00:45:02.810 --> 00:45:12.190 align:middle line:90% 00:45:12.190 --> 00:45:29.160 align:middle line:84% And finally, the set of non-zero eigenvalues of A 00:45:29.160 --> 00:45:36.420 align:middle line:90% is either finite or countable. 00:45:36.420 --> 00:45:47.090 align:middle line:90% 00:45:47.090 --> 00:45:56.480 align:middle line:84% If it is countable, i.e. it's given by a sequence lambda n, 00:45:56.480 --> 00:46:01.330 align:middle line:90% then these-- 00:46:01.330 --> 00:46:04.180 align:middle line:84% or if it is countably infinite, I should say-- 00:46:04.180 --> 00:46:08.720 align:middle line:90% 00:46:08.720 --> 00:46:11.405 align:middle line:84% and I should have said countably infinite here. 00:46:11.405 --> 00:46:17.090 align:middle line:90% 00:46:17.090 --> 00:46:24.050 align:middle line:84% If it's countably infinite, then the eigenvalues converge to 0. 00:46:24.050 --> 00:46:29.300 align:middle line:90% 00:46:29.300 --> 00:46:31.700 align:middle line:84% In particular, this implies that if I 00:46:31.700 --> 00:46:35.840 align:middle line:84% have a compact self-adjoint operator 00:46:35.840 --> 00:46:38.690 align:middle line:84% with infinitely many eigenvalues, 00:46:38.690 --> 00:46:44.090 align:middle line:84% then 0 is in the spectrum of this operator 00:46:44.090 --> 00:46:48.900 align:middle line:84% because the spectrum is a closed set. 00:46:48.900 --> 00:46:51.630 align:middle line:84% So it's closed undertaking limits. 00:46:51.630 --> 00:46:54.270 align:middle line:84% And since, these are in the spectrum, 00:46:54.270 --> 00:46:55.770 align:middle line:84% the limit has to be in the spectrum. 00:46:55.770 --> 00:47:05.330 align:middle line:90% 00:47:05.330 --> 00:47:13.190 align:middle line:90% All right, so proof of 1-- 00:47:13.190 --> 00:47:15.830 align:middle line:84% suppose I have a non-zero eigenvalue. 00:47:15.830 --> 00:47:22.580 align:middle line:84% And towards the contradiction, E lambda 00:47:22.580 --> 00:47:26.390 align:middle line:90% is not finite dimensional. 00:47:26.390 --> 00:47:28.220 align:middle line:90% Then what I can do-- 00:47:28.220 --> 00:47:31.760 align:middle line:84% by the Gram-Schmidt process, then there 00:47:31.760 --> 00:47:45.270 align:middle line:84% exists a sequence or a countable collection un, 00:47:45.270 --> 00:47:54.580 align:middle line:84% orthonormal elements in E lambda. 00:47:54.580 --> 00:47:59.790 align:middle line:84% So every element in the sequence has unit length. 00:47:59.790 --> 00:48:02.790 align:middle line:84% And it's orthogonal to any other element in the sequence. 00:48:02.790 --> 00:48:09.270 align:middle line:90% 00:48:09.270 --> 00:48:23.810 align:middle line:84% Now, since A is a compact operator and all of these 00:48:23.810 --> 00:48:31.280 align:middle line:84% have unit length, it follows that A applied to un 00:48:31.280 --> 00:48:34.620 align:middle line:84% is contained-- this is a sequence in a compact set, 00:48:34.620 --> 00:48:35.120 align:middle line:90% right? 00:48:35.120 --> 00:48:37.290 align:middle line:84% So it has a convergent subsequence-- 00:48:37.290 --> 00:48:44.340 align:middle line:90% 00:48:44.340 --> 00:48:45.195 align:middle line:90% Au nj, j. 00:48:45.195 --> 00:48:56.760 align:middle line:90% 00:48:56.760 --> 00:49:05.260 align:middle line:90% Then Au nj is Cauchy. 00:49:05.260 --> 00:49:07.840 align:middle line:84% But let's actually look at what's 00:49:07.840 --> 00:49:17.240 align:middle line:84% the difference between two of these in norm. 00:49:17.240 --> 00:49:20.150 align:middle line:90% Let's make it squared. 00:49:20.150 --> 00:49:27.470 align:middle line:84% This is equal to norm of, because these are eigenvalues, 00:49:27.470 --> 00:49:33.590 align:middle line:84% lambda un j minus lambda un k squared, which equals-- 00:49:33.590 --> 00:49:39.270 align:middle line:90% 00:49:39.270 --> 00:49:43.178 align:middle line:84% squared, which equals 2 lambda squared, 00:49:43.178 --> 00:49:45.720 align:middle line:84% which is a fixed number that's positive because lambda is not 00:49:45.720 --> 00:49:46.320 align:middle line:90% equal to 0. 00:49:46.320 --> 00:49:51.360 align:middle line:90% 00:49:51.360 --> 00:49:54.300 align:middle line:90% Oh, I left off a part of the-- 00:49:54.300 --> 00:49:56.110 align:middle line:90% OK, so what does this imply? 00:49:56.110 --> 00:50:00.390 align:middle line:84% This implies that the distance between any-- so this is-- 00:50:00.390 --> 00:50:03.222 align:middle line:90% 00:50:03.222 --> 00:50:05.890 align:middle line:84% if I take any two elements in this subsequence, 00:50:05.890 --> 00:50:09.100 align:middle line:84% their distance is a constant equal to 2 times 00:50:09.100 --> 00:50:10.090 align:middle line:90% lambda squared. 00:50:10.090 --> 00:50:11.660 align:middle line:84% And therefore, this is not Cauchy, 00:50:11.660 --> 00:50:12.760 align:middle line:90% which is a contradiction. 00:50:12.760 --> 00:50:26.560 align:middle line:90% 00:50:26.560 --> 00:50:29.068 align:middle line:90% Something I forgot to say-- 00:50:29.068 --> 00:50:30.235 align:middle line:90% restatement of the theorem-- 00:50:30.235 --> 00:50:33.450 align:middle line:90% 00:50:33.450 --> 00:50:36.360 align:middle line:84% forgive me, it's the end of a long day-- 00:50:36.360 --> 00:50:41.250 align:middle line:84% is that eigenvalues have to be real 00:50:41.250 --> 00:50:43.290 align:middle line:84% for self-adjoint compact operators, 00:50:43.290 --> 00:50:47.010 align:middle line:84% or really for self-adjoint operators. 00:50:47.010 --> 00:50:48.510 align:middle line:84% So I could have included it earlier. 00:50:48.510 --> 00:50:51.120 align:middle line:90% 00:50:51.120 --> 00:50:53.880 align:middle line:84% So the eigenvalues of a self-adjoint operator 00:50:53.880 --> 00:50:54.870 align:middle line:90% have to be real. 00:50:54.870 --> 00:50:55.485 align:middle line:90% Why is that? 00:50:55.485 --> 00:51:02.250 align:middle line:90% 00:51:02.250 --> 00:51:12.560 align:middle line:84% Since if I have something with norm 1-- 00:51:12.560 --> 00:51:16.640 align:middle line:84% so if lambda is an eigenvalue, it 00:51:16.640 --> 00:51:19.700 align:middle line:84% comes with an eigenvector u with length 1 00:51:19.700 --> 00:51:21.170 align:middle line:90% so that Au equals lambda u. 00:51:21.170 --> 00:51:23.030 align:middle line:84% Of course, it just has to be a non-zero u. 00:51:23.030 --> 00:51:28.190 align:middle line:84% But I can normalize it by dividing by its length. 00:51:28.190 --> 00:51:30.780 align:middle line:84% And therefore, I get that lambda, 00:51:30.780 --> 00:51:35.930 align:middle line:84% which is equal to lambda u inner product u-- 00:51:35.930 --> 00:51:41.570 align:middle line:84% this is norm of u squared, which is 1 lambda u, u. 00:51:41.570 --> 00:51:47.420 align:middle line:84% And this is equal to complex conjugate-- 00:51:47.420 --> 00:51:49.160 align:middle line:90% or let's not do that. 00:51:49.160 --> 00:51:52.840 align:middle line:90% 00:51:52.840 --> 00:51:58.440 align:middle line:84% This is equal to Au, u which is equal to-- 00:51:58.440 --> 00:52:00.780 align:middle line:84% I take A and it becomes A-star u. 00:52:00.780 --> 00:52:04.680 align:middle line:84% But A-star is equal to A, So I get u, Au, 00:52:04.680 --> 00:52:06.480 align:middle line:90% since A is self-adjoint. 00:52:06.480 --> 00:52:09.990 align:middle line:84% And this is equal to u, lambda u. 00:52:09.990 --> 00:52:15.540 align:middle line:84% And remember, inner products are conjugate linear 00:52:15.540 --> 00:52:16.440 align:middle line:90% in the second entry. 00:52:16.440 --> 00:52:19.930 align:middle line:84% So this lambda pops out, but now complex conjugate. 00:52:19.930 --> 00:52:29.010 align:middle line:84% So lambda-- so we've shown that the complex conjugate is 00:52:29.010 --> 00:52:31.958 align:middle line:90% equal to the original number. 00:52:31.958 --> 00:52:33.375 align:middle line:84% So lambda has to be a real number. 00:52:33.375 --> 00:52:36.060 align:middle line:90% 00:52:36.060 --> 00:52:37.470 align:middle line:84% All, right so that proves part 1, 00:52:37.470 --> 00:52:40.740 align:middle line:84% that the eigenvalues of a self-adjoint operator 00:52:40.740 --> 00:52:42.460 align:middle line:90% have to be real. 00:52:42.460 --> 00:52:48.030 align:middle line:84% And the eigenspaces, which is what I have just 00:52:48.030 --> 00:52:50.760 align:middle line:84% started calling the E lambdas, the eigenspace, 00:52:50.760 --> 00:52:53.640 align:middle line:84% have to be finite dimensional for a compact self-adjoint 00:52:53.640 --> 00:52:54.300 align:middle line:90% operator. 00:52:54.300 --> 00:52:59.890 align:middle line:90% 00:52:59.890 --> 00:53:13.170 align:middle line:84% OK, so now, let's show that distinct eigenspaces have 00:53:13.170 --> 00:53:14.610 align:middle line:90% to be orthogonal to each other. 00:53:14.610 --> 00:53:17.460 align:middle line:84% Suppose lambda 1 does not equal lambda 2. 00:53:17.460 --> 00:53:23.094 align:middle line:90% u1 is in E lambda 1. 00:53:23.094 --> 00:53:26.230 align:middle line:90% u2 is in E lambda 2. 00:53:26.230 --> 00:53:28.810 align:middle line:90% 00:53:28.810 --> 00:53:31.180 align:middle line:84% So now, what I'd like to show is that the inner product 00:53:31.180 --> 00:53:34.632 align:middle line:90% of u1 with u2 is equal to 0. 00:53:34.632 --> 00:53:37.090 align:middle line:84% And it's going to be a trick, kind of like I just did here. 00:53:37.090 --> 00:53:46.660 align:middle line:84% Lambda 1 times u1, u2, this is equal to lambda 1 u1, u2. 00:53:46.660 --> 00:53:51.670 align:middle line:84% This is equal to A applied to u1, u2. 00:53:51.670 --> 00:53:56.740 align:middle line:84% And now, I move A over to here because A is self-adjoint. 00:53:56.740 --> 00:54:04.150 align:middle line:90% 00:54:04.150 --> 00:54:06.310 align:middle line:90% And A applied to u2-- 00:54:06.310 --> 00:54:09.070 align:middle line:84% so u2 is in the second eigenspace. 00:54:09.070 --> 00:54:13.630 align:middle line:84% So this is equal to u1 lambda 2 u2. 00:54:13.630 --> 00:54:17.980 align:middle line:84% And because lambda 1 and lambda 2 have to be real numbers-- 00:54:17.980 --> 00:54:20.530 align:middle line:84% what we've done from the first part-- 00:54:20.530 --> 00:54:24.880 align:middle line:84% this lambda 2 comes all the way out and remains itself, 00:54:24.880 --> 00:54:26.440 align:middle line:84% no complex conjugate because it's 00:54:26.440 --> 00:54:29.020 align:middle line:90% equal to its complex conjugate. 00:54:29.020 --> 00:54:32.920 align:middle line:84% And so I started off with lambda 1 times the inner product 00:54:32.920 --> 00:54:33.940 align:middle line:90% of u1, u2. 00:54:33.940 --> 00:54:38.140 align:middle line:84% And I've ended up with lambda 2 u1 inner product with u2. 00:54:38.140 --> 00:54:41.140 align:middle line:84% And therefore, lambda 1 minus lambda 2 00:54:41.140 --> 00:54:44.560 align:middle line:84% times the inner product of u1 minus-- 00:54:44.560 --> 00:54:47.590 align:middle line:84% or the inner product of u1 with u2 equals 0. 00:54:47.590 --> 00:54:50.350 align:middle line:90% 00:54:50.350 --> 00:54:53.800 align:middle line:84% And lambda 1-- remember, we're assuming lambda 1 00:54:53.800 --> 00:54:56.180 align:middle line:84% and lambda 2 are non-zero-- or not equal. 00:54:56.180 --> 00:54:58.310 align:middle line:84% So this quantity here is non-zero. 00:54:58.310 --> 00:55:04.120 align:middle line:90% So I get that u1, u2 equals 0. 00:55:04.120 --> 00:55:07.460 align:middle line:84% And that's the-- nope, that's not the end. 00:55:07.460 --> 00:55:10.030 align:middle line:84% That's the end of number 2, but not the end 00:55:10.030 --> 00:55:12.745 align:middle line:90% of the proof of this theorem. 00:55:12.745 --> 00:55:15.237 align:middle line:90% 00:55:15.237 --> 00:55:17.320 align:middle line:84% All right, so we're going to prove the last thing, 00:55:17.320 --> 00:55:20.440 align:middle line:84% that the set of non-zero eigenvalues 00:55:20.440 --> 00:55:25.360 align:middle line:84% is either finite or countable, and that if I arrange them 00:55:25.360 --> 00:55:27.925 align:middle line:84% in a sequence, then the sequence converges to 0. 00:55:27.925 --> 00:55:37.360 align:middle line:90% 00:55:37.360 --> 00:55:41.095 align:middle line:84% OK, so just to have some notation running around-- 00:55:41.095 --> 00:55:43.600 align:middle line:90% 00:55:43.600 --> 00:55:48.770 align:middle line:84% capital lambda, let me let this denote 00:55:48.770 --> 00:55:50.630 align:middle line:90% those non-zero eigenvalues. 00:55:50.630 --> 00:56:10.660 align:middle line:90% 00:56:10.660 --> 00:56:14.230 align:middle line:84% All right, so what I'd like to claim-- 00:56:14.230 --> 00:56:29.580 align:middle line:84% or what I'm going to show is that if lambda n is a sequence 00:56:29.580 --> 00:56:36.230 align:middle line:90% of distinct elements-- 00:56:36.230 --> 00:56:54.520 align:middle line:84% or distinct eigenvalues, non-zero eigenvalues of A, 00:56:54.520 --> 00:56:59.380 align:middle line:90% then these converge to 0. 00:56:59.380 --> 00:57:03.020 align:middle line:90% So this gives us-- 00:57:03.020 --> 00:57:06.620 align:middle line:84% of course, so the set of non-zero eigenvalues 00:57:06.620 --> 00:57:08.150 align:middle line:90% may be finite. 00:57:08.150 --> 00:57:08.840 align:middle line:90% Fine. 00:57:08.840 --> 00:57:10.550 align:middle line:90% Suppose it's not, OK? 00:57:10.550 --> 00:57:16.520 align:middle line:84% Now, we're just in the setting that A has 00:57:16.520 --> 00:57:20.510 align:middle line:90% infinitely many eigenvalues. 00:57:20.510 --> 00:57:22.150 align:middle line:84% If I can prove this claim, then I 00:57:22.150 --> 00:57:23.620 align:middle line:90% have proven two things at once. 00:57:23.620 --> 00:57:27.880 align:middle line:84% I have proven both that the set of non-zero eigenvalues 00:57:27.880 --> 00:57:31.480 align:middle line:84% is countably infinite, assuming it's infinite, 00:57:31.480 --> 00:57:34.630 align:middle line:90% and they converge to zero. 00:57:34.630 --> 00:57:35.680 align:middle line:90% So why does this-- 00:57:35.680 --> 00:57:50.890 align:middle line:90% 00:57:50.890 --> 00:57:55.590 align:middle line:84% So first off, if we can show that this capital lambda is 00:57:55.590 --> 00:57:57.720 align:middle line:84% countable, then this claim then implies 00:57:57.720 --> 00:57:59.400 align:middle line:84% that-- or countably infinite, then 00:57:59.400 --> 00:58:02.790 align:middle line:84% this claim tells me that the eigenvalues converge to 0, 00:58:02.790 --> 00:58:04.660 align:middle line:90% which is the last thing I want. 00:58:04.660 --> 00:58:08.670 align:middle line:84% So all I really need to show is that this is countable 00:58:08.670 --> 00:58:09.880 align:middle line:90% using this claim. 00:58:09.880 --> 00:58:14.100 align:middle line:84% Now, why does this show that capital lambda is countable? 00:58:14.100 --> 00:58:19.140 align:middle line:84% Since then if I define lambda sub capital 00:58:19.140 --> 00:58:26.320 align:middle line:84% N to be the set of non-zero eigenvalues 00:58:26.320 --> 00:58:29.200 align:middle line:84% which are, let's say, even bigger than or equal to 1 00:58:29.200 --> 00:58:34.560 align:middle line:84% over N, this has to be a finite set, right? 00:58:34.560 --> 00:58:37.020 align:middle line:84% If it was infinite, then I could find 00:58:37.020 --> 00:58:46.090 align:middle line:84% a sequence of distinct elements in here and obtain-- 00:58:46.090 --> 00:58:49.990 align:middle line:84% or I should say, then I can find a subsequence-- or hold on. 00:58:49.990 --> 00:58:51.970 align:middle line:90% Let me stop for a minute. 00:58:51.970 --> 00:58:59.080 align:middle line:90% 00:58:59.080 --> 00:59:04.080 align:middle line:84% So my claim is that this is finite for all N, which 00:59:04.080 --> 00:59:06.585 align:middle line:84% implies that lambda, which is the union of-- 00:59:06.585 --> 00:59:11.760 align:middle line:90% 00:59:11.760 --> 00:59:14.820 align:middle line:90% is countable. 00:59:14.820 --> 00:59:21.060 align:middle line:84% OK, so assuming this claim or assuming what I wrote here, 00:59:21.060 --> 00:59:24.240 align:middle line:84% that this is finite for all N, this implies this is countable, 00:59:24.240 --> 00:59:25.060 align:middle line:90% that's clear. 00:59:25.060 --> 00:59:27.930 align:middle line:84% So why do I get this as finite, this set 00:59:27.930 --> 00:59:30.790 align:middle line:90% is finite assuming this claim? 00:59:30.790 --> 00:59:34.200 align:middle line:84% Well, if this set is infinite, then I 00:59:34.200 --> 00:59:37.890 align:middle line:84% can pick out a sequence of distinct elements in lambda sub 00:59:37.890 --> 00:59:40.590 align:middle line:90% N that converges. 00:59:40.590 --> 00:59:42.630 align:middle line:84% I could just take any sequence, and then 00:59:42.630 --> 00:59:45.630 align:middle line:84% take a convergent subsequence because that sequence has 00:59:45.630 --> 00:59:50.820 align:middle line:84% to be bounded between 1 over N and the norm of A. 00:59:50.820 --> 00:59:53.340 align:middle line:84% But since they're all bigger than or equal to 1 over N, 00:59:53.340 --> 00:59:56.730 align:middle line:84% that sequence has to converge to something that's non-zero. 00:59:56.730 --> 00:59:58.373 align:middle line:84% But that would contradict the claim-- 00:59:58.373 --> 00:59:59.790 align:middle line:84% again, assuming the claim is true. 00:59:59.790 --> 01:00:01.470 align:middle line:84% We haven't proved it yet, all right? 01:00:01.470 --> 01:00:04.920 align:middle line:84% So again, from this claim, we can then 01:00:04.920 --> 01:00:10.200 align:middle line:84% conclude that each of these sets is finite for all N. 01:00:10.200 --> 01:00:13.530 align:middle line:84% And therefore, the set of non-zero eigenvalues 01:00:13.530 --> 01:00:16.770 align:middle line:90% is countable. 01:00:16.770 --> 01:00:19.350 align:middle line:84% And if it's countably infinite, then, again, from this claim, 01:00:19.350 --> 01:00:23.310 align:middle line:84% we conclude that the eigenvalues must converge to 0 when 01:00:23.310 --> 01:00:25.990 align:middle line:90% I line them up in a sequence. 01:00:25.990 --> 01:00:28.905 align:middle line:84% So the whole proof is reduced to just proving this claim. 01:00:28.905 --> 01:00:37.990 align:middle line:90% 01:00:37.990 --> 01:00:56.670 align:middle line:84% OK, so to prove the claim, let un be associated eigenvectors. 01:00:56.670 --> 01:01:11.260 align:middle line:90% 01:01:11.260 --> 01:01:15.070 align:middle line:90% So these have unit lengths. 01:01:15.070 --> 01:01:21.670 align:middle line:84% And for all n, A un equals lambda n un, right? 01:01:21.670 --> 01:01:22.700 align:middle line:90% We have eigenvalues. 01:01:22.700 --> 01:01:25.360 align:middle line:84% So we can find eigenvectors with unit length. 01:01:25.360 --> 01:01:31.440 align:middle line:90% 01:01:31.440 --> 01:01:40.850 align:middle line:84% Now, then lambda n, which is equal to-- 01:01:40.850 --> 01:01:42.530 align:middle line:84% or the absolute value of lambda n 01:01:42.530 --> 01:01:46.460 align:middle line:84% is equal to the absolute value of lambda n-- 01:01:46.460 --> 01:01:49.670 align:middle line:84% or the norm of lambda n applied to-- 01:01:49.670 --> 01:01:54.710 align:middle line:84% or times un, which is equal to the norm of A applied to un. 01:01:54.710 --> 01:01:57.380 align:middle line:90% 01:01:57.380 --> 01:02:00.050 align:middle line:84% So what I'm going to show is that A 01:02:00.050 --> 01:02:02.210 align:middle line:90% applied to un converges to 0. 01:02:02.210 --> 01:02:07.160 align:middle line:84% So if you like, this is the final claim that I need to. 01:02:07.160 --> 01:02:08.780 align:middle line:90% Prove so this is claim 1. 01:02:08.780 --> 01:02:16.020 align:middle line:90% 01:02:16.020 --> 01:02:19.530 align:middle line:84% Claim 1 will follow from claim 2 in this little computation 01:02:19.530 --> 01:02:25.800 align:middle line:84% right here, where claim 2 is that the norm of A 01:02:25.800 --> 01:02:29.910 align:middle line:84% applied to un-- again, un's are eigenvectors with unit length 01:02:29.910 --> 01:02:32.040 align:middle line:84% corresponding to the lambda n's converge to 0. 01:02:32.040 --> 01:02:40.010 align:middle line:90% 01:02:40.010 --> 01:02:44.660 align:middle line:84% So the fact that A applied to these unit vectors converges 01:02:44.660 --> 01:02:51.070 align:middle line:84% to 0 is not just specific to eigenvectors 01:02:51.070 --> 01:02:53.050 align:middle line:90% of distinct eigenvalues. 01:02:53.050 --> 01:02:56.980 align:middle line:84% It's just a property of the compactness of A and the fact 01:02:56.980 --> 01:03:00.250 align:middle line:84% that the un's are in orthonormal sequence. 01:03:00.250 --> 01:03:01.510 align:middle line:90% They're all unit length. 01:03:01.510 --> 01:03:03.790 align:middle line:84% And any one element in the sequence 01:03:03.790 --> 01:03:07.750 align:middle line:84% is orthogonal to a different element in the sequence. 01:03:07.750 --> 01:03:08.890 align:middle line:90% So suppose not. 01:03:08.890 --> 01:03:10.330 align:middle line:90% Suppose claim 2 does not hold. 01:03:10.330 --> 01:03:16.160 align:middle line:90% 01:03:16.160 --> 01:03:21.260 align:middle line:84% Then just negating the definition of convergence, 01:03:21.260 --> 01:03:23.060 align:middle line:84% there exists an epsilon positive. 01:03:23.060 --> 01:03:27.020 align:middle line:90% 01:03:27.020 --> 01:03:45.500 align:middle line:84% And we can find a subsequence A unj such that for all j, 01:03:45.500 --> 01:03:49.940 align:middle line:84% length of A unj is bigger than or equal to epsilon 0. 01:03:49.940 --> 01:03:53.720 align:middle line:84% If you look at the definition of-- 01:03:53.720 --> 01:03:55.910 align:middle line:84% or the definition of convergence to 0 and then 01:03:55.910 --> 01:04:00.050 align:middle line:84% negate that, you can conclude that you can find a subsequence 01:04:00.050 --> 01:04:01.070 align:middle line:90% so that I have this. 01:04:01.070 --> 01:04:03.800 align:middle line:84% So there is some bad epsilon 0 so that I have that. 01:04:03.800 --> 01:04:07.530 align:middle line:90% 01:04:07.530 --> 01:04:16.570 align:middle line:84% All right, since A is a compact operator, 01:04:16.570 --> 01:04:19.920 align:middle line:84% there exists a further subsequence. 01:04:19.920 --> 01:04:26.360 align:middle line:90% 01:04:26.360 --> 01:04:31.250 align:middle line:84% And let me call it e sub k, which is un sub j sub k-- 01:04:31.250 --> 01:04:37.330 align:middle line:90% 01:04:37.330 --> 01:04:41.030 align:middle line:90% unj-- such that-- 01:04:41.030 --> 01:04:45.760 align:middle line:84% so remember, A applied to un sub j-- so un sub 01:04:45.760 --> 01:04:47.290 align:middle line:90% j is a unit length vector. 01:04:47.290 --> 01:04:50.620 align:middle line:84% And therefore, A applied to that is contained in a compact set, 01:04:50.620 --> 01:04:52.490 align:middle line:84% assuming A is a compact operator. 01:04:52.490 --> 01:04:55.780 align:middle line:84% So this must have a convergent subsequence such 01:04:55.780 --> 01:05:09.000 align:middle line:84% that A applied to ek converges in H. 01:05:09.000 --> 01:05:16.800 align:middle line:84% And note Aek, since this is just a subsequence of this sequence, 01:05:16.800 --> 01:05:20.160 align:middle line:84% is bigger than or equal to epsilon 0 for all k. 01:05:20.160 --> 01:05:23.440 align:middle line:90% 01:05:23.440 --> 01:05:26.620 align:middle line:84% Now, since the ek's are a subsequence 01:05:26.620 --> 01:05:31.240 align:middle line:84% of an orthonormal sequence, it's still an orthonormal sequence. 01:05:31.240 --> 01:05:41.220 align:middle line:84% So note, for all k not equal to l inner product ek el, 01:05:41.220 --> 01:05:50.640 align:middle line:90% which is unk unl equals 0. 01:05:50.640 --> 01:05:52.600 align:middle line:90% And what I'm using here-- 01:05:52.600 --> 01:05:54.720 align:middle line:84% so of course, these are all unit vectors. 01:05:54.720 --> 01:05:56.190 align:middle line:90% Why are they orthogonal? 01:05:56.190 --> 01:06:00.300 align:middle line:84% It's because they correspond to distinct eigenvalues, 01:06:00.300 --> 01:06:02.340 align:middle line:90% distinct non-zero eigenvalues. 01:06:02.340 --> 01:06:05.670 align:middle line:84% And we proved that-- that was number 2, that-- 01:06:05.670 --> 01:06:07.940 align:middle line:90% was it number 2? 01:06:07.940 --> 01:06:12.200 align:middle line:84% Yeah, that eigenvectors for distinct eigenvalues 01:06:12.200 --> 01:06:15.760 align:middle line:90% are orthogonal to each other. 01:06:15.760 --> 01:06:19.240 align:middle line:84% So assuming the negation of the claim 2, 01:06:19.240 --> 01:06:20.740 align:middle line:84% which would prove claim 1 and finish 01:06:20.740 --> 01:06:23.080 align:middle line:84% the proof of this whole theorem, I 01:06:23.080 --> 01:06:28.150 align:middle line:84% conclude that there exists a sequence of eigenvectors, 01:06:28.150 --> 01:06:34.770 align:middle line:84% orthonormal eigenvectors of A So that Aek is always bounded 01:06:34.770 --> 01:06:36.390 align:middle line:90% below in norm by epsilon 0. 01:06:36.390 --> 01:06:43.980 align:middle line:90% 01:06:43.980 --> 01:06:46.780 align:middle line:90% And this sequence converges. 01:06:46.780 --> 01:06:56.940 align:middle line:84% So let f be the limit as k goes to infinity of Aek. 01:06:56.940 --> 01:07:02.430 align:middle line:84% Then norm of f, by continuity of the norm, 01:07:02.430 --> 01:07:05.490 align:middle line:84% is equal to the limit of the norms of the ek's. 01:07:05.490 --> 01:07:08.010 align:middle line:90% And all of these are above-- 01:07:08.010 --> 01:07:11.310 align:middle line:84% bigger than or equal to epsilon 0. 01:07:11.310 --> 01:07:13.680 align:middle line:90% So f is non-zero, right? 01:07:13.680 --> 01:07:22.482 align:middle line:90% 01:07:22.482 --> 01:07:24.065 align:middle line:84% In fact, we can say a little bit more. 01:07:24.065 --> 01:07:27.620 align:middle line:90% 01:07:27.620 --> 01:07:30.890 align:middle line:90% Then in fact-- let's see. 01:07:30.890 --> 01:07:34.170 align:middle line:90% 01:07:34.170 --> 01:07:37.620 align:middle line:84% So this is kind of useless information I skipped. 01:07:37.620 --> 01:07:40.418 align:middle line:90% 01:07:40.418 --> 01:07:41.960 align:middle line:84% I didn't write down what I wanted to. 01:07:41.960 --> 01:07:46.280 align:middle line:84% But then-- well no, I still need that. 01:07:46.280 --> 01:07:49.530 align:middle line:90% No, let me not get rid of that. 01:07:49.530 --> 01:07:50.940 align:middle line:90% So that should still be there. 01:07:50.940 --> 01:07:53.630 align:middle line:84% So the norm of f is bigger than or equal to epsilon 0. 01:07:53.630 --> 01:07:56.360 align:middle line:84% So norm of f squared is bigger than or equal 01:07:56.360 --> 01:07:57.575 align:middle line:90% to epsilon 0 squared. 01:07:57.575 --> 01:08:01.960 align:middle line:90% 01:08:01.960 --> 01:08:04.290 align:middle line:90% So f inner product f-- 01:08:04.290 --> 01:08:10.140 align:middle line:84% and by continuity of the inner product, that's equal to-- 01:08:10.140 --> 01:08:14.940 align:middle line:84% since the Aek's converts to f, I will get f here. 01:08:14.940 --> 01:08:20.399 align:middle line:84% And using the fact that A is self-adjoint, 01:08:20.399 --> 01:08:22.124 align:middle line:90% this is equal to ek, Af. 01:08:22.124 --> 01:08:27.830 align:middle line:90% 01:08:27.830 --> 01:08:39.310 align:middle line:84% So I have that this limit here is non-negative. 01:08:39.310 --> 01:08:40.540 align:middle line:90% I mean, it's a real number. 01:08:40.540 --> 01:08:51.520 align:middle line:84% And it's bigger than or equal to epsilon 0 squared. 01:08:51.520 --> 01:08:53.310 align:middle line:90% Now, here's the problem. 01:08:53.310 --> 01:08:57.090 align:middle line:84% I have here a sequence of orthonormal vectors, right? 01:08:57.090 --> 01:08:59.609 align:middle line:84% And I know that the sum of squares of these Fourier 01:08:59.609 --> 01:09:02.160 align:middle line:84% coefficients, which are Fourier coefficients for A 01:09:02.160 --> 01:09:05.160 align:middle line:90% applied to f, are-- 01:09:05.160 --> 01:09:06.840 align:middle line:90% the sum of squares is finite. 01:09:06.840 --> 01:09:09.090 align:middle line:84% And therefore, this has to go to 0. 01:09:09.090 --> 01:09:12.580 align:middle line:84% And that's the contradiction to the epsilon 0 squared. 01:09:12.580 --> 01:09:19.920 align:middle line:84% So by Bessel, Bessel's inequality, 01:09:19.920 --> 01:09:27.450 align:middle line:84% we get that sum over k norm ek, Af squared, 01:09:27.450 --> 01:09:31.529 align:middle line:84% this is less than or equal to the norm of Af squared, which 01:09:31.529 --> 01:09:34.010 align:middle line:90% is finite. 01:09:34.010 --> 01:09:37.970 align:middle line:84% And since this is a convergent series, 01:09:37.970 --> 01:09:40.460 align:middle line:84% the individual terms have to converge to 0. 01:09:40.460 --> 01:09:44.220 align:middle line:90% 01:09:44.220 --> 01:09:50.479 align:middle line:90% And therefore, this equals 0. 01:09:50.479 --> 01:09:56.361 align:middle line:84% But this and this are a contradiction. 01:09:56.361 --> 01:10:04.550 align:middle line:90% 01:10:04.550 --> 01:10:16.780 align:middle line:84% OK, so that finishes the proof of this theorem 01:10:16.780 --> 01:10:22.330 align:middle line:84% about the eigenvalues and eigenspaces 01:10:22.330 --> 01:10:24.280 align:middle line:84% for a compact self-adjoint operator. 01:10:24.280 --> 01:10:38.910 align:middle line:90% 01:10:38.910 --> 01:10:43.960 align:middle line:84% All right, so I think we'll stop there. 01:10:43.960 --> 01:10:51.000 align:middle line:90%