Selfadjoint operators in spaces of functions of infinitely many variables:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Math. Soc.
1986
|
| Series: | Translations of mathematical monographs
63 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | EST: Samosoprjažennye operatory v prostranstvach funkcij beskonečnogo čisla peremennych <engl.>. - Aus d. Russ. übers. |
| Physical Description: | XV, 383 S. |
| ISBN: | 0821845152 |
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| 240 | 1 | 0 | |a Samosoprjažennye operatory v prostranstvach funkcij bezkonečnogo čisla peremennych |
| 245 | 1 | 0 | |a Selfadjoint operators in spaces of functions of infinitely many variables |c by Yu. M. Berezanskii |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 1986 | |
| 300 | |a XV, 383 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Translations of mathematical monographs |v 63 | |
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Record in the Search Index
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| adam_text | Contents
Foreword to the English translation ix
Foreword xi
Introduction 1
CHAPTER 1. Spaces of Generalized Functions 9
§1. The concept of a space with negative norm 10
1. Definition of the negative space 10
2. The Hilbert property of the negative space 12
3. Coincidence of the negative space with the dual of the
positive space 12
4. Construction of isometries between the spaces in the chain 13
5. Adjointness with respect to the zero space 14
6. Construction of a chain from a subspace of the positive space 16
7. Construction of a chain from the negative space 16
8. Construction of a chain from an operator 18
9. Weighted orthogonal sums of chains 18
10. Projective limits of Banach spaces 19
§2. Finite and infinite tensor products of Hilbert spaces 23
1. Tensor products of finitely many Hilbert spaces 23
2. Tensor products of finitely many operators 25
3. Tensor products of infinitely many Hilbert spaces 27
4. Imbeddings of tensor products 30
5. Tensor products of chains 33
6. Weighted tensor products of Hilbert spaces 36
7. Tensor products of infinitely many operators 40
8. The kernel theorem 43
9. The case of bilinear forms. Positive definite kernels 49
10. Complete von Neumann tensor product of infinitely many
Hilbert spaces 51
11. Triplets of spaces 57
iii
iv CONTENTS
§3. Spaces of functions of finitely many variables 60
1. Spaces of square integrable functions with a weight 60
2. Sobolev spaces on a bounded domain 60
3. The delta function as an element of a negative Sobolev space 62
4. Quasinuclearity of imbeddings of Sobolev spaces 63
5. The kernel theorem in spaces of square integrable functions
on bounded domains 66
6. The spaces Wl2 69
7. Sobolev spaces on an unbounded domain 70
8. Sobolev spaces with a weight 73
9. The Schwartz space S(RN) as a projective limit of Sobolev
spaces 75
10. The space D(RN) as a projective limit of Sobolev spaces 77
11. The kernel theorem in spaces of square integrable functions 79
12. The chain constructed from a continuous positive definite
kernel 83
§4. Spaces of functions of infinitely many variables 86
1. The space of square integrable functions with respect to a
product measure and the rigging of it 86
2. The case of a Gaussian measure 87
3. Weighted infinite tensor products of nuclear spaces 92
4. Examples of infinite tensor products of nuclear spaces 95
5. The space A{R°°) 98
6. The space #(R°°) 107
CHAPTER 2. General Selfadjoint and Normal Operators 113
§1. A joint resolution of the identity 117
1. A general resolution of the identity 117
2. Products of finitely many resolutions of the identity 119
3. Construction of a joint resolution of the identity in the
general case 121
4. Topologization 122
5. Regularity of a joint resolution of the identity 124
6. The concept of the support of a measure and its properties 125
7. Families of multiplication operators 128
8. Construction of a measure on a larger space from a measure
on a smaller space. Compactification 130
9. Construction of a measure on a smaller space from a measure
on a larger space. Modification of a measure 132
10. Properness of a joint resolution of the identity 135
11. Spectral representation of a family of commuting normal
operators 138
CONTENTS v
§2. The spectral theorem 139
1. Differentiation of an operator valued measure with respect to
its trace 139
2. Differentiation of a resolution of the identity. The spectral
measure 144
3. Differentiation of a joint resolution of the identity 146
4. The case of a nuclear rigging 147
5. The concept of a joint eigenvector and the spectrum of a
family of operators 148
6. The spectral theorem. The case of at most countably many
operators 149
7. The spectral theorem. The general case 151
8. The spectral theorem. The case of at most countably many
unbounded operators 163
9. Continuity and smoothness of eigenvalues 164
10. Supplementary remarks 164
11. The Fourier transformation. The direct integral of Hilbert
spaces 165
§3. Connections with random processes and commutative algebras of
operators. Quantum processes 170
1. The concept of a random process 170
2. Construction of a family of commuting normal operators
from a random process 171
3. Construction of a random process from a family of
commuting normal operators. Connection with commutative
algebras of operators 173
4. The concept of a quantum process 181
§4. Representation of algebraic structure by commuting operators 183
1. Representations of a group 183
2. Generalization of Stone s theorem 184
3. Representations of a semigroup 193
4. Representations of a linear space 193
5. Generalized quantum processes 195
6. Representations of an algebra 196
7. Representations of more general structures 197
§5. Supplementary results on expansions in generalized eigenvectors 198
1. A converse theorem 199
2. The case of an imbedding that is not quasinuclear 200
3. Expansion in eigenfunctions of Carleman operators 203
4. Expansion in generalized eigenfunctions of operators acting in
Z 2 spaces 211
5. Existence of a rigging 213
vi CONTENTS
6. Connection with the theory of commutative normed algebras
and the nuclear spectral theorem 219
§6. Selfadjointness of operators and uniqueness of the solution of the
Cauchy problem for evolution equations 225
1 General theorems on the connection between selfadjointness
and uniqueness 225
2. Some generalizations 231
3. Essential selfadjointness and quasianalytic vectors 237
4. Verification of commutativity for selfadjoint operators 241
5. Parabolic criteria for selfadjointness 246
6. Selfadjointness of perturbations of an operator by a potential 248
Chapter 3. Some Classes of Operators Acting in a Space of Functions
of Infinitely Many Variables 255
§1. Operators acting with respect to different variables and functions
of them 258
1. Operators acting with respect to different variables 258
2. The Fourier Wiener transformation 264
3. Functions of operators acting with respect to different
variables 268
4. The case of functions which can be well approximated by
cylindrical functions 272
§2. Operators admitting separation of infinitely many variables 276
1. The convolution of finitely many resolutions of the identity
on the line 276
2. The convolution of infinitely many resolutions of the identity
on the line and its existence 285
3. Separation of infinitely many variables 294
§3. Differential operators with infinitely many variables 302
1. Some facts about general operators with separating variables 302
2. Differential operators with infinitely many separating
variables 306
3. Differential operators with infinitely many separating
variables. The case of annihilation 309
4. First digression. Selfadjointness of differential operators in
the case of finitely many variables 313
5. Continuation of the digression: Getting rid of the additional
smoothness restrictions on the leading coefficients 322
6. Selfadjointness of differential operators with infintely many
variables. The parabolic approach 325
7. Selfadjointness of differential operators with infinitely many
variables. Combination of the parabolic approach and the
hyperbolic approach 335
8. Second digression. Operators that are not semibounded below 337
CONTENTS vii
Comments on the literature 343
Bibliography 357
Subject index 381
|
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| author_sort | Berezanskij, Jurij M. |
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| id | DE-604.BV000719665 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:18:21Z |
| institution | BVB |
| isbn | 0821845152 |
| language | English |
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| physical | XV, 383 S. |
| publishDate | 1986 |
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| publisher | American Math. Soc. |
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| series | Translations of mathematical monographs |
| series2 | Translations of mathematical monographs |
| spelling | Berezanskij, Jurij M. Verfasser aut Samosoprjažennye operatory v prostranstvach funkcij bezkonečnogo čisla peremennych Selfadjoint operators in spaces of functions of infinitely many variables by Yu. M. Berezanskii Providence, RI American Math. Soc. 1986 XV, 383 S. txt rdacontent n rdamedia nc rdacarrier Translations of mathematical monographs 63 EST: Samosoprjažennye operatory v prostranstvach funkcij beskonečnogo čisla peremennych <engl.>. - Aus d. Russ. übers. Distribution Funktionalanalysis (DE-588)4070505-5 gnd rswk-swf Spektraltheorie (DE-588)4116561-5 gnd rswk-swf Distribution Funktionalanalysis (DE-588)4070505-5 s Spektraltheorie (DE-588)4116561-5 s DE-604 Translations of mathematical monographs 63 (DE-604)BV000002394 63 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000449846&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Berezanskij, Jurij M. Selfadjoint operators in spaces of functions of infinitely many variables Translations of mathematical monographs Distribution Funktionalanalysis (DE-588)4070505-5 gnd Spektraltheorie (DE-588)4116561-5 gnd |
| subject_GND | (DE-588)4070505-5 (DE-588)4116561-5 |
| title | Selfadjoint operators in spaces of functions of infinitely many variables |
| title_alt | Samosoprjažennye operatory v prostranstvach funkcij bezkonečnogo čisla peremennych |
| title_auth | Selfadjoint operators in spaces of functions of infinitely many variables |
| title_exact_search | Selfadjoint operators in spaces of functions of infinitely many variables |
| title_full | Selfadjoint operators in spaces of functions of infinitely many variables by Yu. M. Berezanskii |
| title_fullStr | Selfadjoint operators in spaces of functions of infinitely many variables by Yu. M. Berezanskii |
| title_full_unstemmed | Selfadjoint operators in spaces of functions of infinitely many variables by Yu. M. Berezanskii |
| title_short | Selfadjoint operators in spaces of functions of infinitely many variables |
| title_sort | selfadjoint operators in spaces of functions of infinitely many variables |
| topic | Distribution Funktionalanalysis (DE-588)4070505-5 gnd Spektraltheorie (DE-588)4116561-5 gnd |
| topic_facet | Distribution Funktionalanalysis Spektraltheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000449846&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000002394 |
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