Arveson W. — A Short Course on Spectral Theory :: Электронная библиотека попечительского совета мехмата МГУ

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Arveson W. — A Short Course on Spectral Theory

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Название: A Short Course on Spectral Theory

Автор: Arveson W.

Аннотация:

This book presents the basic tools of modern analysis within the context of the fundamental problem of operator theory: to calculate spectra of specific operators on infinite dimensional spaces, especially operators on Hilbert spaces. The tools are diverse, and they provide the basis for more refined methods that allow one to approach problems that go well beyond the computation of spectra: the mathematical foundations of quantum physics, noncommutative k-theory, and the classification of simple C*-algebras being three areas of current research activity which require mastery of the material presented here. The book is based on a fifteen-week course which the author offered to first or second year graduate students with a foundation in measure theory and elementary functional analysis.

Язык:

Рубрика: Математика/Анализ/Функциональный анализ/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2002

Количество страниц: 135

Добавлена в каталог: 09.04.2005

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Предметный указатель

$A^{-1}$       11M
      110
      108
$H^{\infty}$       108
      12
      8
$\mathcal{F}(E)$       93
$\mathcal{T}$       110
$\sigma (A)$       6
$\sigma$ -representation      60
$\sigma(T)$       64
$\sigma_w(T)$       96
Absolutely convergent seria      8 11
Adjoint of an operator      40
Algebra, Banach      8
Algebra, complex      7
Algebra, disk      8
Algebra, division      17
Algebra, group      10
Algebra, matrix      8
Algebra, normed      8
Asymptotic invariant      83 85
Atkinson’s theorem      93
Banach $\sigma$ -algebra      57
Banach algebra      8
Banach limit      85
Basis for a vector space      24
Beurling’s theorem      118
Beurling’s theorem, proof of      120
C(X)      8
C-algebra of operators      42
Calkin algebra      83
Carona of $\mathbb{R}$       81
Closed convex bull      28
Coburn’s theorem      119
Cokernel      87
Commutant of a set of operators      43
Commutative C-algebras, characterization of      47
Compact operator      13 68
Convex bull      28
Coordinate systems and unitary operators      52
Current variable      31 64 69 105 100 112
Curve      34
CYCLE      35
Cyclic representation      59
Diagonalizable operator      53
Direct sum of representations      57
Disk algebra      8
Division algebra      17
Equivalent representations      58
Essential range of $f\in L^{\infty}$       44
Essential representation      57
Essential spectrum      94
Exact sequence      21
Exponential map      47
Extension, split      112
Extension, Toeplita      112
F. and M. Riesz theorem      118
Finite-rank operator      13
Fredbolm index      95
Fredholm alternative      87
Fredholm operator on a Banach space      93
Free Abelian group      35
Functional calculus, analytic      33 37
Functional calculus, Borel      63
Functional calculus, continuous      51
Gelfand spectrum      28
Gelfand spectrum, compactness of      25
Gelfand — Mazur theorem      19
Gelfand — Naimark theorem      129
Gelland map      26
General linear group      14
General linear group, openness      14
GNS construction      122
GNS pair for states on Banach $\sigma$ -algebras      123
GNS pair, exlatance      133
GNS pair, inequality      123
Group algebra      10
Haar measure      10
Hartman theorems      122
Hausdorff maximality principle      23
Hilbert — Schmidt operator      70
Hold of $K\in \mathbb{C}$       32
Ideal      21
Ideal in C-algebra      70
Ideal, numerical      23
Ideal, proper      21
Index of a linear transformation      80
Index of a product      97
Index of Toeplitz operators      116
Index, continuity of      99
Index, of a bounded operator      95
Index, stability of      68
Integral equations, Volterra      4
Interior of a cycle      36
Invertible element of a Banach algebra      14
Invertible operator      6
Involution      41
Isometry      41
Isomorphism of Banach algebras      17
K(E)      80
Kernel      87
Linearly ordered set      23
Linearly ordered subset, maximal      29
lnductive partially ordered set      23
Locally analytic function      30
MASA      102
Maximal abelian von Neumann algebra      102
Maximal element      23
Maximal ideal      23

Maximal ideal space      26
Maximal ideal, closure or      23
Multiplication algebra      44
Multiplication operator      43
Multiplicity-free operator      103
Neumann series      7 14
Nondegenerate representation      67
Normal operator      41
Numerical radius      45
Numerical range      45
Oriented curve      34
Partially ordered set      23
Polarization formula for linear forms      45
Polarization formula for operators      72 75
Positive linear functional      123
Positive operator      41
Projection      41
Proper ideal      21
Pure isometry      113
Pure state      120
Quasinilpotent operator      20
Quotient, algebra      21
Quotient, Banach algebra      22
Quotient, C-algebra      79
Quotient, norm      22
R(X)      19
Radical      27
Radon measure      10
Range of a *-homomorphism      80
Rank of an operator      13
Reducing subspace      113
Regular representation      13
Representation, cyclle      50
Representation, direct sum      57
Representation, essential space of      67
Representation, nondegenerale      67
Representation, norm of      68
Representation, of a Banach *-algebra      87
Representation, subrepresentation of      58
Resolution of the identity      67
Resolvent, estimates      14
Resolvent, set      6
Riesz lemma for operators      40
Riesz lemma for vectors      30
Runge’s theorem      33
Schwarz inequality for positive linear functionals      123
SCROC      34
Semisimple      27
Separable C*-algebra      105 129
Separable measure space      43
Set, inductive      23
Set, linearly ordered      23
Set, partially ordered      23
Shift, bilateral      104
Shift, unilateral      110
Shift, weighted      18
Simple algebra      21
Simple topologically      22 24
sp(A)      25
Spectral mapping theorem      19
Spectral measure      66
Spectral permanence for Banach algebras      32
Spectral permanence, C*-algebras      49
Spectral radius      19
Spectral radius formula      19
Spectral theorem      65
Spectrum and Gelland theorem      36
Spectrum and solving equations      2
Spectrum and the complex number field      2
Spectrum in a Banach algebra      10
Spectrum of a compact operator      87
Spectrum of a multiplication operetor      44
Spectrum of a Toeplitz operator      121 123
Spectrum of an operator      6
Spectrum, compactness      10
Spectrum, Gelfend      25
Spectrum, nontriviality      16
State      122
State, pure      129
Stone — Cech compactification of X      81
Subrepresentation      58
Symbol of a Toeplitz matrix      109
Symbol of Toeplitz operator      107
Tauberian theorems      29
Toeplitz C*-algebra      110
Toeplitz matrix      101 107
Toeplitz operator      101 107
Toeplitz operator, characterization of      107
Toeplitz operator, index of      110
Toeplitz operator, spectrum of      121 122
Topology, locally convex      42
Topology, strong operator topology      42
Topology, weak operator topology      42
Trace class operator      71
Unilateral shift      110
Unit of an algebra      7
Unit, approximate      9
Unital algebra      7
Unitarily equivalent representations      58
Unitary operator      41
von Neumann algebra      42 102
Weighted shift      18
Weyl spectrum      93
Widow’s theorem      121
Wiener algebra      29
Winding number of a curve      35
Winding number of a cycle      36
Winding number of an element or $C(T)^{-1}$       115
Wold decomposition      113
Zorn’s Lemma      23

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