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Gohberg I., Goldberg S., Kaashoek M. — Classes of linear operators. (volume 1)

Gohberg I., Goldberg S., Kaashoek M. — Classes of linear operators. (volume 1)

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Название: Classes of linear operators. (volume 1)

Авторы: Gohberg I., Goldberg S., Kaashoek M.

Аннотация:

This book presents a panorama of operator theory. It treats a variety of classes of linear operators which illustrate the richness of the theory, both in its theoretical developments and its applications. For each of the classes various differential and integral operators motivate or illustrate the main results. The topics have been updated and enhanced by new developments, many of which appear here for the first time. Interconnections appear frequently and unexpectedly.
This second volume consists of five parts: triangular representations, classes of Toeplitz operators, contractive operators and characteristic operator functions, Banach algebras and algebras of operators, and extension and completion problems. The exposition is self-contained and has been simplified and polished in an effort to make advanced topics accessible to a wide audience of students and researchers in mathematics, science and engineering.

Publication date: September 1993 Of interest to: Researchers, graduate students Codes: M16002, M16126, M16088, M16096 Classes of Linear Operators Vol. I This book presents a panorama of operator theory. It treats a variety of classes of linear operators which illustrate the richness of the theory, both in its theoretical developments and its applications. For each of the classes various differential and integral operators motivate or illustrate the main results. The topics have been updated and enhanced by new developments, many of which appear here for the first time. Interconnections appear frequently and unexpectedly.

The present volume consists of four parts: general spectral theory, classes of compact operators, Fredholm and Wiener-Hopf operators, and classes of unbounded operators: The exposition is self-contained and has been simplified and polished in an effort to make advanced topics accessible to a wide audience of students and researchers in mathematics, science and engineering.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1990

Количество страниц: 468

Добавлена в каталог: 16.06.2005

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Предметный указатель

$AC_{n}(J)$ a space of functions on the interval J      295
$A_{\Im}$ imaginary part of the operator A      135 146
$A_{\Re}$ real part of the operator A      146
$A{\geq}0$ non-negative operator A      82
$C^{1}$ -solution      408
$C^{k}(\Omega)$ a set of differentiate functions on $\Omega$       310
$C^{k}(\overline{\Omega})$ a set of differentiable functions on $\overline{\Omega}$       310
$C^{\infty}(\Omega)$ the set of $C^{\infty}$ -functions on $\Omega$       310
$C^{\infty}(\Omega)$ the set of $C^{\infty}$ -functions with compact support in ${\Omega}$       310
$C_{0}$ semigroup      415
$D^{\alpha}$ elementary differential operator      310
$E_{A}$ a space spanned by eigenvectors and generalized eigenvectors      31
$H^{0}_{m}(\Omega)$ closure of $C^{\infty}_{0}(\Omega)$ in $H_{m}(\Omega)$       313
$H_{m}(\Omega)$ Sobolev space of order m      313
$I_{X}$ identity operator on X      5
$L^{m}_{2}([a,b])$ Lebesgue space of ${\mathbb{C}}^{m}$ -valued functions      148
$P_{\sigma}(A)$ Riesz projection      9
$S^{\perp}$ orthogonal complement of S, and also at times, annihilator of S      292
$S_{1}$ the trace class operators      105
$S_{2}$ the Hilbert — Schmidt operators      143
$s_{j}$ (A) j-th singular value of the operator A      96 212
$T_{max,{\tau},J}$ maximal operator      295
$T_{min,{\tau},J}$ minimal operator      300
$T_{R,J}$ a differential operator      300
$W_{A}$ numerical range of A      167
$\emptyset$ empty set      143
$\Im_{\lambda}$ the imaginary part of the complex number $\lambda$       5
$\mathbb{C}$ field of complex numbers      5
$\mathbf{D}$ open unit disk in the complex plane      310
$\mathcal{D}(A)$ domain of the operator A      288
$\mathcal{K}(X)$ the set of compact operators on the Banach space X      191
$\mathcal{L}$ (X) Banach space of all bounded linear operators on X      5
$\mathcal{L}$ (X,Y) Banach space of all bounded linear operators from X into Y      5
$\mathfrak{F}(A)$ a family of analytic functions associated with a bounded operator A      13
$\mathfrak{F}_{\infty}(A)$ a family of analytic functions associated with an unbounded operator A      323
$\mathrm{T}$ unit circle in the complex plane      191 373
$\nu$ (A) number of non-zero singular values of A      96
$\overline{A}$ minimal closed linear extension of the operator A      289
$\rho$ (A) resolvent set of the operator A      5
$\rho$ (G,A) resolvent set of the pencil $\lambda$ G - A      49
$\sigma$ (A) spectrum of the operator A      5
$\sigma$ (G, A) spectrum of the operator pencil $\lambda$ G — A      49
$\sigma_{ess}$ (A) essential spectrum of the operator A      191 373
$\|A\|_{1}$ trace class norm of A      106
$\|A\|_{2}$ Hilbert — Schmidt norm of A      143
${ }^{\perp}N$ inverse annihilator of N      292
${(x_{1},... ,x_{n})}^{T}$ transpose of the row vector ( $x_{1},... ,x_{n}$ )      292
${\lambda}_{j}$ (A) j-th eigenvalue of the compact operator A      30
${\mathbb{C}}^{n}$ complex Euclidean n-space      5
${\mathbb{C}}_{\infty}$ the extended complex plane      49
${\mathbf{R}}^{n}$ real Euclidean n-space      9
${\mathcal{R}}^{m{\times}m}_{\infty}(R)$ a set of rational ${m{\times}m}$ matrix functions      232
${\partial{\Omega}}$ boundary of the open set $\Omega$       315
${\Re}{\lambda}$ real part of the complex number A      9
< $\cdot,\cdot$ > product      292
A' conjugate of the operator A      292
A(X ${\rightarrow}$ Y) operator A with domain in X and range in Y      288
A* (Hilbert space) adjoint of the operator A      291
a-invariant      326
Abstract Cauchy problem      408
Adjoint boundary conditions      306
Adjoint of an unbounded operator      290
Adjoint of differential operators      301 321 383
Adjoint, Lagrange adjoint      301 309 318
Algebraic multiplicity of an eigenvalue for operator functions      205
Algebraic multiplicity of an eigenvalue for operators      26
Algebraic multiplicity of an eigenvalue relative to a contour      205
Analytic at a point      6
Analytic on a set      6
Analytically matricially coupled      46
Associated operator      54
Asymptotically stable, difference equation      58
Asymptotically stable, differential equation      20
Asymptotically stable, solution      20
A|N restriction of A to the subspace of N      8 326
Bijection, associated with an operator      188
Boundary of class $C^{2}$       315
Bounded resolution of identity      72
C([a,b]) the space of complex-valued continuous functions on [a,b]      288
Calkin algebra      191
Canonical factorization      235
Cauchy contour      6
Cauchy domain      6
Cayley transform      346
Closable operator      289
Closed linear extension      289
Closed linear extension, minimal      289
Compact semigroup      435
Complete system of eigenvectors and generalized, eigenvectors      135 378
Completeness problem of      29
Conjugate of an unbounded operator      292
Continuously differentiable solution      408
Contraction semigroup      427
Convolution operator with symbol      216
Coupling matrices      45
Coupling relation      45
d(A) codimension of the range of A      184
Derivative      7
Derivative, weak derivative      312
det A determinant of an operator matrix A      194
det(I + A) determinant of I + A with A a trace class operator      117
Determinant      114
Determinant, analyticity of      119
Differentiable      7
Differential operator      295 316 382 399
Differential operator, adjoint of      301 321 383
Differential operator, higher order      399
Differential operator, on a half line      382
Differential operator, selfadjoint      341
Differential operator, uniformly elliptic      316
dim M dimension of the linear space M      117
Dirichlet operator      316
Dirichlet operator, adjoint of      321
Dissipative operator on a Banach space      429
Dissipative operator on a Hilbert space      428
Divergence form      316
Domain, Cauchy domain      6
Domain, inner domain      49
Domain, outer domain      49
E(t) orthogonal projection in a resolution of the identity      72
Eigenvalue of an unbounded operator      349
Eigenvalue of finite type for a bounded operator      25
Eigenvalue of finite type for an operator function      205
Eigenvalue of finite type for an unbounded operator      326
Eigenvector of an unbounded operator      349
Eigenvector of an unbounded operator, generalized      378
Elementary solutions      29
Equivalent, globally equivalent      38
Equivalent, locally equivalent      42
Equivalent, strictly equivalent      237
Essential spectrum for a bounded operator      191
Essential spectrum for an unbounded operator      373
Exponential representation      268
Exponential type zero      122
Extension      38
Extension of an unbounded operator      290
Factorization, canonical      235
Factorization, indices      235
Factorization, left Wiener — Hopf      235
Factorization, right Wiener — Hopf      235 236
Factorization, Wiener — Hopf      234
Final space of a partial isometry      84
Finite slab problem      274
Finitely meromorphic      206
First kind Wiener — Hopf operator      398
Fourier transform      217
Fourier transformation      217
Fredholm operator, bounded      184
Fredholm operator, perturbation of      189 376
Fredholm operator, unbounded      372

Functional calculus      13 80 323
Fundamental matrix      154
G(A) graph of the operator A      289
Generalized Dirichlet problem      316
Generalized eigenspace      30
Generalized eigenvectors      28 378
Generalized inverse      192
Generalized inverse in the weak sense      193
Generalized resolvent equation      50
Generator of a semigroup      419
Geometric multiplicity      26
Globally equivalent      38
Graph norm      369
Green's formula      304
Green's function      386
Half range problem      258
Hankel operator      225
Hardy space      222
Hilbert — Schmidt norm      143
Hilbert — Schmidt operator      140
Hille — Yoshida — Phillips theorem      419
I identity operator      5
Im A range (image) of the operator A      5
ind(A) index of the operator A      184
Index of a bounded Fredholm operator      184
Index of a bounded semi-Fredholm operator      283
Index of a unbounded Fredholm operator      372
Indicator      151
Infinitesimal generator of a semigroup      419
Initial space of a partial isometry      84
Inner domain      49
Integral operator      148
Invariant, projectionally      181
Invariant, subspace      8
Isolated part of the spectrum      9
Isometric semigroup      432
Jordan chain      28
k( $\Gamma$ ; 0) winding number      226
Ker A null space of A      226
Kernel function      148 215
Kernel function, semi-separable      149
Lagrange adjoint for an ordinary differential expression      301 309
Lagrange adjoint for the Dirichlet problem      318
Laplace transform      418
Lax — Milgram lemma      319
Left strictly equivalent      237
Left Wiener — Hopf factorization      235
Lemma of Schur      31
Linear operator pencil      49
Locally equivalent      42
Lower order      149
Lyapunov's theorem      22
m( ${\Gamma},W({\cdot})$ ) algebraic multiplicity relative to a contour $\Gamma$       205
m( ${\lambda}_{0}$ ,A) algebraic multiplicity      26
m( ${\lambda}_{0},W({\cdot})$ ) algebraic multiplicity of an operator function      205
Matricially coupled      44
Matricially coupled, analytically      46
Maximal operator      295
Maximal operator, adjoint of      301
Min-max theorem      96
Minimal closed linear extension      289
Minimal operator      300
Minimal operator, adjoint of      301
Multiplication, operator of      218
Multiplicity, algebraic multiplicity      26 205
Multiplicity, geometric multiplicity      26
Multiplying medium      275
Mutually disjoint projections      88
n(A) dimension of the null space of A      184
Non-negative operator      82
Normal with respect to a contour      205
Numerical range      167
Open neighbourhood of a set      13
Operator of multiplication      218
Operator with compact resolvent      328
Operator, bounded selfadjoint      61
Operator, closable      289
Operator, convolution      216
Operator, differential      295 382 399
Operator, Dirichlet      316
Operator, dissipative      428 429
Operator, Fredholm      184 372
Operator, Hilbert — Schmidt      140
Operator, non-negative      82
Operator, semi-Fredholm      283
Operator, strictly negative      22
Operator, strictly positive      22
Operator, trace class      105
Operator, unbounded selfadjoint      341
Operator, unitary      85
Operator, Volterra      32
Operator, Wiener — Hopf      221
Order of a semiseparable representation      149
Order of a semiseparable representation, lower      149
Order of a semiseparable representation, upper      149
Order of a Sobolev space      313
Order of the differential expression $D^{\alpha}$       310
Outer domain      49
Partial isometry      84
Partial isometry, final space of      84
Partial isometry, initial space of      84
Polar decomposition      84
Problem of completeness      29
Projectionally invariant      181
rank A rank of the operator A      232
Realization      244
Realized form      244 395
Regular      49
Regular, $\Delta$ -regular      49
Resolution of the identity      72 350 362
Resolution of the identity for a bounded selfadjoint operator      74
Resolution of the identity for an unbounded selfadjoint operator      362
Resolution of the identity, bounded      72
Resolvent equation      9 288
Resolvent equation, generalized      50
Resolvent identity      288
Resolvent operator      6 288
Resolvent set      5 49 288
Restriction of an unbounded operator      290
Restriction to an invariant subspace      8
Riesz projection      9 326
Right canonical factorization      235
Right equivalence operator      54
Right factorization indices      235
Right Wiener — Hopf factorization, relative to circle      236
Right Wiener — Hopf factorization, relative to the real line      235
Row multiplicity      238
S-number      96 212
Scattering function      258
Schmidt-representation      97
Schur basis      136
Schur's lemma      31
Selfadjoint operator      61 341
Semi-Fredholm operator      283
Semi-separable kernel function      149
Semi-separable representation      149
Semi-separable representation, lower order of      149
Semi-separable representation, order of      149
Semi-separable representation, upper order of      149
Semigroup, $C_{0}$ semigroup      415
Semigroup, compact      435
Semigroup, contraction      427
Semigroup, isometric      432
Semigroup, strongly continuous      415
Semigroup, translation      420
Semigroup, unitary      432
Separating projection      54
Sesquilinear form      319
Simple pole      88
Singular value      96 212

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