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 there’s 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 a

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 say 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 W 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 invert 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 CN

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 a

142 00:03:43,169 –> 00:03:44,389 set of all the E

143 00:03:44,580 –> 00:03:45,199 values.

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 W

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 nontrivial 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 the 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 a

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.