Obata N. — White Noise Calculus and Fock Space :: Электронная библиотека попечительского совета мехмата МГУ

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Obata N. — White Noise Calculus and Fock Space

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Название: White Noise Calculus and Fock Space

Автор: Obata N.

Аннотация:

White Noise Calculus is a distribution theory on Gaussian space, proposed by T. Hida in 1975. This approach enables us to use pointwise defined creation and annihilation operators as well as the well-established theory of nuclear space.This self-contained monograph presents, for the first time, a systematic introduction to operator theory on fock space by means of white noise calculus. The goal is a comprehensive account of general expansion theory of Fock space operators and its applications. In particular,first order differential operators, Laplacians, rotation group, Fourier transform and their interrelations are discussed in detail w.r.t. harmonic analysis on Gaussian space. The mathematical formalism used here is based on distribution theory and functional analysis , prior knowledge of white noise calculus is not required.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 2008

Количество страниц: 193

Добавлена в каталог: 04.07.2008

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Предметный указатель

$(E^{\otimes(l+m)}_{\mathds{C}})^{*}_{sym(l,m)}$       83
$(\mathfrak{X}^{\otimes n})^{*}_{sym}$       17
$:x^{\otimes n}:$       25
$d\Gamma(T)$       107
$d_{Y}$       71
$GL(\mathfrak{X})$       118
$H_{n}$       168
$p_{\xi}$ , $q_{\xi}$       145
$S\Phi$       49
$s_{l,m}(\kappa)$       82
$S_{\lambda}$       104
$t_{m,l}(\kappa)$       84
$T_{y}$       76
$\cong$       1
$\Delta$       33
$\Delta_{G}$       121
$\Gamma(a)$       34
$\Gamma(T)$       107
$\Gamma(\mathfrak{H})$       30
$\mathcal{B}(\mathfrak{X},\mathfrak{N})$ , $\mathcal{B}(\mathfrak{X},\mathfrak{N};\mathfrak{Z})$       10
$\mathcal{B}_{sep}(\mathfrak{X},\mathfrak{N})$ , $\mathcal{B}_{sep}(\mathfrak{X},\mathfrak{N};\mathfrak{Z})$       11
$\mathcal{H}_{n}(\mathds{R})$ , $\mathcal{H}_{n}(\mathds{C})$       28
$\mathcal{L}(\mathfrak{X},\mathfrak{N})$       9
$\mathcal{P}(\mathds{R})$ , $\mathcal{P}(\mathds{C})$       24
$\mathcal{P}_{n}(\mathds{R})$ , $\mathcal{P}_{n}(\mathds{C})$       23
$\mathcal{Q}_{n}(\mathds{R})$ , $\mathcal{Q}_{n}(\mathds{C})$       25
$\mathcal{S}_{A}(\Omega)$ , $\mathcal{S}^{*}_{A}(\Omega)$       11
$\mathfrak{B}$       16
$\mathfrak{F}$ , $\mathfrak{F}_{\theta}$       140
$\mathfrak{G}_{\theta}$       140
$\mathfrak{S}_{n}$       17
$\mathfrak{X}^{*}$       2
$\mu$ , $\mu_{\sigma}$       20
$\nabla$       109
$\otimes^{l}$ , $\otimes_{l}$       54—55
$\otimes_{\pi}$       9
$\partial_{t}$       72
$\phi_{\xi}$       31 48
$\rho$       33
$\tau$       24
$\tau_{m}$       102
$\widehat{\otimes}$       17
$\widehat{\otimes}_{l}$       57
$\widehat{\Xi}$       88
$\widetilde{\Xi}$       88
$\Xi_{l,m}(\kappa)$       82
$|\cdot|_{l,m;p,q}$       54
(E*, $\mu$ )      20
Annihilation operator      73
Bilinear map, jointly continuous      10
Bilinear map, separately continuous      11
Bochner — Minlos theorem      17
Boson Fock space      see "Fock space"
Canonical commutation relation (CCR)      75 87 156
Canonical commutation relation (CCR), Weyl form      149
CH-space      3
CH-space, standard      5
Characteristic function      16
Characterization theorem for generalized functionals      65
Characterization theorem for operator symbols      97
Characterization theorem for operator symbols on vector-valued functionals      166
Characterization theorem for test functionals      65
Characterization theorem for vector-valued functionals      162
Coherent state      see "Exponential vector"
Continuous version theorem      38
Continuous version theorem for vector-valued functional      161
Contraction of tensor product      55
Contraction of tensor product, left      55
Contraction of tensor product, norm estimates      169
Contraction of tensor product, right      55
Contraction of tensor product, vector-valued version      163
Countably Hilbert space      see "CH-space"
Creation operator      73
Cylindrical $\sigma$ -field      16
Defining seminorms      1
Defining seminorms, directed family of      1
Derivation      114
Differential second quantization      107
Dual space      see "Strong dual space"
e(i), e(j), e(k)      53—54
Equicontinuous subset      3
Exponential vector      31 48
First order differential operator      112
First order differential operator with constant coefficients      114
First order differential operator with smooth coefficients      114
Fock expansion      98
Fock expansion of operators on vector-valued functional      166
Fock space      30
Fock space, Guichardet's realization      152
Fourier transform      140
Fourier transform of probability measures      17
Fourier transform, characterization      147
Fourier transform, Fock expansion      142
Fourier transform, relation with S-transform      143
Fourier — Mehler transform      140 159
Fourier — Mehler transform, characterization      147
Fourier — Mehler transform, Fock expansion      142
Frechet space      3

Gaussian measure      19
Gaussian measure, rotation invariance      126
Gaussian measure, translation quasi-invariance      23
Gaussian space      19
Gelfand triple      19
Gradient operator      109
Gross Laplacian      121
Gross Laplacian in discrete coordinate      123
Hermite polynomial      168
Hida's differential operator      72
Hilbertian seminorm      3
Hypotheses (H1)-(H3)      11
Infinite dimensional rotation group      126
Infinitesimal generator      118
Infinitesimal generator of adjoint Fourier — Mehler transforms      142
Infinitesimal generator of rotations      127
Infinitesimal generator of scaling operators      124
Infinitesimal generator of translations      120
Integral kernel operator      82
Integral kernel operator on vector-valued functionals      164
Kernel distribution      82
Kernel theorem      11
Kuo's Fourier transform      see "Fourier transform"
Levy Laplacian      150
Locally convex space      1
Locally convex space, bounded subset      2
Locally convex space, direct product      1
Locally convex space, direct sum      1
Locally convex space, inductive limit      2
Locally convex space, inductive system      2
Locally convex space, projective limit      1
Locally convex space, projective system      1
Locally convex space, reduced projective system      2
Locally convex space, reflexive      3
Multiplication operator      63
Multiplication operator, Fock expansion      102
N      121
Nuclear space      7
Number operator      121
Number operator in discrete coordinate      123
O(E;H)      126
One-parameter subgroup      118
One-parameter subgroup of Fourier — Mehler transforms      140
One-parameter subgroup of scaling operators      124
One-parameter subgroup of translations      120
One-parameter subgroup, differentiable      118
One-parameter subgroup, regular      120
Operator-valued distribution      162
Polarization formula      167
Polynomial on Gaussian space      24
Polynomial on Gaussian space, wick-ordered      26
Rotation-invariant distribution      134
Rotation-invariant operators      132
S-transform      49
S-transform of vector-valued functionals      161
Scaling operator      104
Scaling operator, Fock expansion      105
Schroedinger representation      157
Second quantization      107
Strong dual space      2
Strong dual space, defining Hilbertian seminorms      41
Symbol      88
Symbol of operators on vector-valued functionals      165
Symmetric Hilbert space      see "Fock space"
T-transform      143
Taylor expansion      101
Tensor product, $\pi$ -      9
Tensor product, Hilbert space      9
Tensor product, symmetrization      17
Topology of bibounded convergence      10
Topology of bounded convergence      2 9
Topology, $\pi$ -      9
Topology, Mackey      3
Topology, strong dual      2
Trace      24
Translation operator      76
Translation operator, Fock expansion      100
U((E); ( $L^{2}$ ))      126
U-functional      69
Union-partition identity      152
White noise      38
White noise analogue of Euclidean norm      123 136
White noise as multiplication operator      74 103
White noise delta function      47
White noise functional, generalized      35
White noise functional, test      35
White noise functional, vector-valued      159
White noise polynomial      38
Wick-ordered polynomial      26
Wiener formula      59
Wiener product      62
Wiener — Ito decomposition      29
Wiener — Ito decomposition for vector-valued functionals      160
Wiener — Ito expansion      30 47
Wiener — Ito expansion of generalized functionals      38
Wiener — Ito expansion of vector-valued functionals      160—161
Wiener — Ito — Segal isomorphism      30
x(t)      38 73

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