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Title: Spectrum of Bounded Operators
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Series: Functional Analysis
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YouTube-Title: Functional Analysis 28 | Spectrum of Bounded Operators
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Subtitle in English
1 00:00:00,540 –> 00:00:02,200 Hello and welcome back to
2 00:00:02,210 –> 00:00:03,730 functional analysis
3 00:00:04,280 –> 00:00:05,489 and as always many,
4 00:00:05,500 –> 00:00:06,889 many thanks to all the nice
5 00:00:06,900 –> 00:00:07,949 people that support this
6 00:00:07,960 –> 00:00:09,779 channel on Steady or PayPal.
7 00:00:10,329 –> 00:00:10,729 Today
8 00:00:10,739 –> 00:00:12,520 in part 28 we will talk
9 00:00:12,529 –> 00:00:14,220 about the spectrum of a bounded
10 00:00:14,229 –> 00:00:14,880 operator.
11 00:00:15,619 –> 00:00:17,049 The spectrum comes in as
12 00:00:17,059 –> 00:00:18,639 a generalization for the
13 00:00:18,909 –> 00:00:20,200 eigenvalues of a matrix.
14 00:00:20,370 –> 00:00:21,229 For this
15 00:00:21,239 –> 00:00:22,840 please recall when we have
16 00:00:22,850 –> 00:00:24,209 a square matrix A,
17 00:00:24,770 –> 00:00:26,469 which means we have n rows
18 00:00:26,479 –> 00:00:28,440 and n columns and the entries
19 00:00:28,450 –> 00:00:29,760 can come from the complex
20 00:00:29,770 –> 00:00:30,319 numbers,
21 00:00:31,059 –> 00:00:32,580 then we are able to talk
22 00:00:32,590 –> 00:00:34,069 about the eigenvalues of
23 00:00:34,080 –> 00:00:34,290 A.
24 00:00:34,959 –> 00:00:36,330 In particular, we call a
25 00:00:36,340 –> 00:00:38,290 complex number lambda an
26 00:00:38,319 –> 00:00:39,119 eigenvalue,
27 00:00:39,840 –> 00:00:41,669 if we find a corresponding
28 00:00:41,680 –> 00:00:43,419 eigenvector. More
29 00:00:43,430 –> 00:00:44,900 concretely, this means there
30 00:00:44,909 –> 00:00:46,779 exists a vector x
31 00:00:46,790 –> 00:00:48,509 which is not the zero vector
32 00:00:49,150 –> 00:00:50,580 and it fulfills that
33 00:00:50,590 –> 00:00:52,340 Ax is equal to
34 00:00:52,349 –> 00:00:53,130 lambda x.
35 00:00:53,909 –> 00:00:55,270 In other words, the matrix
36 00:00:55,279 –> 00:00:56,750 multiplication for this vector
37 00:00:56,759 –> 00:00:58,750 x is reduced to a scalar
38 00:00:58,759 –> 00:01:00,549 multiplication at this
39 00:01:00,560 –> 00:01:01,029 point.
40 00:01:01,040 –> 00:01:02,790 It’s a good idea to rewrite
41 00:01:02,799 –> 00:01:03,819 this equation.
42 00:01:04,379 –> 00:01:05,790 For example, we can just
43 00:01:05,800 –> 00:01:07,360 bring lambda x to the left-
44 00:01:07,370 –> 00:01:08,989 hand side by using the
45 00:01:09,000 –> 00:01:10,099 identity matrix.
46 00:01:10,559 –> 00:01:12,050 Now we have a new matrix
47 00:01:12,059 –> 00:01:13,669 that sends this vector x
48 00:01:13,680 –> 00:01:14,910 to the zero vector.
49 00:01:15,360 –> 00:01:16,889 However, this then means
50 00:01:16,900 –> 00:01:18,669 that the kernel of this matrix
51 00:01:18,680 –> 00:01:20,370 contains more than just the
52 00:01:20,379 –> 00:01:21,209 zero vector.
53 00:01:21,769 –> 00:01:23,379 Please recall in the kernel
54 00:01:23,389 –> 00:01:24,900 we find all the vectors that
55 00:01:24,910 –> 00:01:26,050 are sent to zero.
56 00:01:26,449 –> 00:01:27,919 Of course, we can also see
57 00:01:27,930 –> 00:01:29,440 this matrix as a map.
58 00:01:29,449 –> 00:01:31,069 So a map that sends a vector
59 00:01:31,080 –> 00:01:33,050 x to the vector A
60 00:01:33,230 –> 00:01:34,099 minus lambda
61 00:01:34,169 –> 00:01:35,309 Ix.
62 00:01:36,190 –> 00:01:37,410 Now having the kernel be
63 00:01:37,419 –> 00:01:38,930 bigger than the zero space
64 00:01:38,940 –> 00:01:40,690 is equivalent to saying that
65 00:01:40,699 –> 00:01:41,970 this map is not
66 00:01:41,980 –> 00:01:42,709 injective.
67 00:01:43,339 –> 00:01:43,730 OK.
68 00:01:43,739 –> 00:01:44,839 This might be a good time
69 00:01:44,849 –> 00:01:46,500 to refresh your linear algebra
70 00:01:46,510 –> 00:01:47,910 knowledge and talk about
71 00:01:47,919 –> 00:01:49,529 the rank nullity theorem.
72 00:01:49,919 –> 00:01:51,290 It holds for all matrices
73 00:01:51,300 –> 00:01:52,879 M where the important thing
74 00:01:52,889 –> 00:01:54,120 is that we have the number
75 00:01:54,129 –> 00:01:55,410 n for the columns.
76 00:01:56,019 –> 00:01:57,370 This number n is the
77 00:01:57,379 –> 00:01:58,809 dimension we have as an
78 00:01:58,819 –> 00:02:00,559 input for this map here
79 00:02:01,220 –> 00:02:02,680 and in the following sense,
80 00:02:02,690 –> 00:02:03,930 this dimension is
81 00:02:03,940 –> 00:02:04,830 conserved.
82 00:02:05,489 –> 00:02:06,709 The new dimension we get
83 00:02:06,720 –> 00:02:08,110 out on the right-hand side
84 00:02:08,119 –> 00:02:09,500 is given by the dimension
85 00:02:09,508 –> 00:02:10,669 of the range of M.
86 00:02:11,270 –> 00:02:12,710 Therefore, this number can’t
87 00:02:12,720 –> 00:02:13,750 be bigger than n
88 00:02:13,860 –> 00:02:15,539 and in the case, it is less
89 00:02:15,550 –> 00:02:16,910 everything else has to go
90 00:02:16,919 –> 00:02:18,190 into the kernel of M.
91 00:02:18,720 –> 00:02:20,429 In other words, both dimensions
92 00:02:20,440 –> 00:02:22,190 have to add up to the original
93 00:02:22,199 –> 00:02:23,470 dimension we put in.
94 00:02:24,039 –> 00:02:25,440 Now because this formula
95 00:02:25,449 –> 00:02:27,000 connects the range and the
96 00:02:27,009 –> 00:02:28,699 kernel, we immediately get
97 00:02:28,710 –> 00:02:30,179 for square matrices
98 00:02:30,190 –> 00:02:31,660 and this map, that
99 00:02:31,669 –> 00:02:32,910 injectivity
100 00:02:32,979 –> 00:02:34,929 bijectivity and surjectivity
101 00:02:34,940 –> 00:02:36,380 are indeed the same thing.
102 00:02:36,889 –> 00:02:38,119 Hence, here, we could also
103 00:02:38,130 –> 00:02:39,699 say this map is not
104 00:02:39,710 –> 00:02:41,410 surjective or simply the
105 00:02:41,419 –> 00:02:43,009 map is not invertible.
106 00:02:43,660 –> 00:02:45,119 However, if we leave the
107 00:02:45,130 –> 00:02:46,750 finite dimensional case,
108 00:02:46,759 –> 00:02:48,710 this rank nullity theorem will
109 00:02:48,720 –> 00:02:49,910 not hold any more.
110 00:02:50,520 –> 00:02:52,179 For this reason, we immediately
111 00:02:52,190 –> 00:02:53,800 get different possibilities
112 00:02:53,809 –> 00:02:55,679 for which the invertibility of this
113 00:02:55,690 –> 00:02:56,899 map can fail.
114 00:02:57,350 –> 00:02:58,449 Now, for the rest of the
115 00:02:58,460 –> 00:02:59,860 video, let X be a
116 00:02:59,869 –> 00:03:01,419 complex Banach space
117 00:03:02,070 –> 00:03:03,919 and T from X to X
118 00:03:03,929 –> 00:03:05,570 should be a bounded linear
119 00:03:05,580 –> 00:03:06,339 operator.
120 00:03:06,910 –> 00:03:07,440 To put it
121 00:03:07,449 –> 00:03:08,740 in other words, X is the
122 00:03:08,750 –> 00:03:10,550 generalization of C^n
123 00:03:10,669 –> 00:03:12,500 and T for the matrix A.
124 00:03:13,169 –> 00:03:14,419 Therefore, the spectrum of
125 00:03:14,429 –> 00:03:16,289 T should be the generalization
126 00:03:16,300 –> 00:03:17,610 of the set of all
127 00:03:17,619 –> 00:03:18,529 eigenvalues.
128 00:03:19,199 –> 00:03:20,669 So it should be a subset
129 00:03:20,679 –> 00:03:22,389 of the complex numbers
130 00:03:22,929 –> 00:03:24,589 and usually it’s denoted
131 00:03:24,600 –> 00:03:26,250 by the lower case Sigma.
132 00:03:27,029 –> 00:03:28,850 Now inside this set Sigma
133 00:03:28,860 –> 00:03:30,330 T we have all the
134 00:03:30,339 –> 00:03:31,970 complex numbers lambda
135 00:03:32,020 –> 00:03:33,690 such that T minus
136 00:03:33,699 –> 00:03:35,529 Lambda identity is
137 00:03:35,539 –> 00:03:36,630 not bijective.
138 00:03:37,179 –> 00:03:38,440 Therefore, if we consider
139 00:03:38,449 –> 00:03:39,820 a finite dimensional vector
140 00:03:39,830 –> 00:03:41,539 space X, we are in this
141 00:03:41,550 –> 00:03:43,160 case again and get out the
142 00:03:43,169 –> 00:03:43,389 set of all the
143 00:03:43,580 –> 00:03:45,199 eigenvalues.
144 00:03:45,940 –> 00:03:47,080 However, for the infinite
145 00:03:47,100 –> 00:03:48,240 dimensional case, we will
146 00:03:48,250 –> 00:03:49,729 see that we can split up
147 00:03:49,740 –> 00:03:51,630 this set into three parts.
148 00:03:52,009 –> 00:03:53,199 Before we do that, let’s
149 00:03:53,210 –> 00:03:54,770 also define the so-called
150 00:03:54,779 –> 00:03:56,389 resolvent set of T
151 00:03:56,949 –> 00:03:58,720 and this one is denoted by
152 00:03:58,729 –> 00:03:59,789 a lower case rho.
153 00:04:00,830 –> 00:04:02,389 The set looks very similar,
154 00:04:02,399 –> 00:04:03,850 but now we look at all the
155 00:04:03,860 –> 00:04:05,789 complex numbers lambda where
156 00:04:05,800 –> 00:04:07,729 this map is indeed bijective
157 00:04:07,889 –> 00:04:09,600 and the inverse is bounded.
158 00:04:10,270 –> 00:04:11,740 So in some sense, these are
159 00:04:11,750 –> 00:04:13,550 the good points, because there
160 00:04:13,559 –> 00:04:15,130 we can invert our bounded
161 00:04:15,139 –> 00:04:15,850 operator.
162 00:04:16,178 –> 00:04:17,529 Of course, at this point,
163 00:04:17,540 –> 00:04:18,829 you know a lot of functional
164 00:04:18,839 –> 00:04:19,608 analysis
165 00:04:19,619 –> 00:04:21,220 and therefore, you see we
166 00:04:21,230 –> 00:04:23,029 are working in a Banach space
167 00:04:23,040 –> 00:04:24,399 and therefore we can use
168 00:04:24,410 –> 00:04:26,209 the bounded inverse theorem,
169 00:04:26,660 –> 00:04:28,279 which simply means when we
170 00:04:28,290 –> 00:04:30,089 have the bijectivity this
171 00:04:30,100 –> 00:04:31,250 immediately follows.
172 00:04:31,690 –> 00:04:32,929 So we can just say
173 00:04:32,940 –> 00:04:34,829 Sigma is the complement
174 00:04:34,839 –> 00:04:35,440 set of
175 00:04:35,450 –> 00:04:37,440 rho. With this, you see
176 00:04:37,450 –> 00:04:38,959 why we need to work in Banach
177 00:04:38,970 –> 00:04:40,579 spaces, because only
178 00:04:40,589 –> 00:04:41,890 there we get out the
179 00:04:41,899 –> 00:04:43,399 inverses as bounded
180 00:04:43,410 –> 00:04:45,179 operators and we
181 00:04:45,190 –> 00:04:46,420 work with complex vector
182 00:04:46,429 –> 00:04:48,140 spaces, because as we will
183 00:04:48,149 –> 00:04:49,459 later see, this spectrum
184 00:04:49,470 –> 00:04:51,070 gives us more information
185 00:04:51,079 –> 00:04:51,839 in this case.
186 00:04:52,399 –> 00:04:53,809 However, of course, all the
187 00:04:53,820 –> 00:04:55,480 definitions here also work
188 00:04:55,489 –> 00:04:57,070 with real vector spaces.
189 00:04:57,079 –> 00:04:58,970 When you substitute C with
190 00:04:58,980 –> 00:05:00,600 R. Knowing all
191 00:05:00,609 –> 00:05:02,149 this, I can show you now
192 00:05:02,160 –> 00:05:03,899 how we can split up the set
193 00:05:03,910 –> 00:05:04,750 sigma T.
194 00:05:05,579 –> 00:05:07,260 The first one is the so-called
195 00:05:07,269 –> 00:05:08,720 point spectrum of T.
196 00:05:09,640 –> 00:05:10,970 Indeed, this is the only
197 00:05:10,980 –> 00:05:12,269 set we have for the finite
198 00:05:12,399 –> 00:05:13,359 dimensional case.
199 00:05:14,010 –> 00:05:15,260 However, in the infinite
200 00:05:15,269 –> 00:05:16,660 dimensional case, we also
201 00:05:16,670 –> 00:05:17,820 have a set, we call the
202 00:05:17,829 –> 00:05:19,640 continuous spectrum and a
203 00:05:19,649 –> 00:05:21,089 set we call the residual
204 00:05:21,100 –> 00:05:21,750 spectrum.
205 00:05:22,320 –> 00:05:23,390 Now, from the discussion
206 00:05:23,399 –> 00:05:24,589 above, you might already
207 00:05:24,600 –> 00:05:26,250 guess that we can split up
208 00:05:26,260 –> 00:05:28,059 the bijectivity here into
209 00:05:28,070 –> 00:05:29,510 injectivity and
210 00:05:29,619 –> 00:05:30,420 surjectivity.
211 00:05:31,149 –> 00:05:32,350 In fact, that’s what we can
212 00:05:32,359 –> 00:05:32,790 do
213 00:05:32,799 –> 00:05:33,890 and in the case that this
214 00:05:33,899 –> 00:05:35,630 operator is not injective,
215 00:05:35,640 –> 00:05:37,309 we define the point spectrum
216 00:05:37,320 –> 00:05:37,820 of T.
217 00:05:38,440 –> 00:05:39,739 Please recall we learned
218 00:05:39,750 –> 00:05:41,279 above that not injective
219 00:05:41,290 –> 00:05:42,730 means this operator has a
220 00:05:42,739 –> 00:05:44,640 non-trivial kernel which
221 00:05:44,649 –> 00:05:46,440 means we have eigenvectors.
222 00:05:47,250 –> 00:05:48,649 In this sense, these points
223 00:05:48,660 –> 00:05:50,130 are indeed the classical
224 00:05:50,230 –> 00:05:51,109 eigenvalues.
225 00:05:51,690 –> 00:05:51,970 OK.
226 00:05:51,980 –> 00:05:53,209 Now, you should see to get
227 00:05:53,220 –> 00:05:54,769 a disjoint union, we also
228 00:05:54,779 –> 00:05:56,029 have to include the injectivity
229 00:05:56,040 –> 00:05:57,070 here.
230 00:05:57,619 –> 00:05:58,820 So in this sense, we could
231 00:05:58,829 –> 00:06:00,450 actually do it, but it turns
232 00:06:00,459 –> 00:06:02,089 out we can distinguish the
233 00:06:02,100 –> 00:06:03,290 points even more.
234 00:06:03,940 –> 00:06:05,609 Now, not surjective, simply
235 00:06:05,619 –> 00:06:07,049 means that the range of the
236 00:06:07,059 –> 00:06:08,910 operator is not the whole
237 00:06:08,920 –> 00:06:09,989 space X.
238 00:06:10,619 –> 00:06:11,940 However, it would be a nice
239 00:06:11,950 –> 00:06:13,709 property to have almost the
240 00:06:13,720 –> 00:06:14,679 space X
241 00:06:15,329 –> 00:06:16,519 and this would mean that
242 00:06:16,529 –> 00:06:18,220 the closure of this set is
243 00:06:18,230 –> 00:06:18,739 X.
244 00:06:19,820 –> 00:06:21,559 Now these points lambda form
245 00:06:21,570 –> 00:06:23,149 the continuous spectrum by
246 00:06:23,160 –> 00:06:25,100 definition. Both names
247 00:06:25,109 –> 00:06:26,630 are chosen in this way, because
248 00:06:26,640 –> 00:06:28,170 for important examples, the
249 00:06:28,179 –> 00:06:29,750 point spectrum consists of
250 00:06:29,760 –> 00:06:31,489 individual points in C
251 00:06:31,540 –> 00:06:33,029 and the continuous spectrum
252 00:06:33,040 –> 00:06:34,609 forms whole intervals.
253 00:06:35,160 –> 00:06:36,579 This also explains the last
254 00:06:36,589 –> 00:06:37,070 name.
255 00:06:37,079 –> 00:06:38,480 The residual spectrum just
256 00:06:38,489 –> 00:06:39,750 gets all other points.
257 00:06:40,549 –> 00:06:40,929 Here
258 00:06:40,940 –> 00:06:42,290 the operator is injective,
259 00:06:42,299 –> 00:06:43,429 but not surjective.
260 00:06:43,570 –> 00:06:45,010 And even the closure of the
261 00:06:45,019 –> 00:06:46,670 range is not X.
262 00:06:47,149 –> 00:06:48,290 Here, I can tell you for
263 00:06:48,299 –> 00:06:49,880 the property that the closure
264 00:06:49,890 –> 00:06:51,200 is the whole set X,
265 00:06:51,209 –> 00:06:52,959 we simply say the range
266 00:06:52,970 –> 00:06:54,450 lies dense in X.
267 00:06:54,950 –> 00:06:56,089 Later, you will see that
268 00:06:56,100 –> 00:06:57,350 for many important
269 00:06:57,359 –> 00:06:58,970 examples, the last set is
270 00:06:58,980 –> 00:06:59,929 indeed empty.
271 00:07:00,589 –> 00:07:01,959 This is not always the case,
272 00:07:01,970 –> 00:07:03,480 but for these examples, we
273 00:07:03,489 –> 00:07:04,989 only have to deal with these
274 00:07:05,000 –> 00:07:05,829 two sets here.
275 00:07:06,540 –> 00:07:06,959 OK.
276 00:07:06,970 –> 00:07:08,500 Then let’s use the next video
277 00:07:08,510 –> 00:07:10,160 to look at some examples.
278 00:07:10,709 –> 00:07:11,989 Therefore, I hope I see you
279 00:07:12,000 –> 00:07:13,480 there and have a nice day.
280 00:07:13,609 –> 00:07:14,239 Bye.
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