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E-raamat: Classical Summation in Commutative and Noncommutative Lp-Spaces

  • Formaat: PDF+DRM
  • Sari: Lecture Notes in Mathematics 2021
  • Ilmumisaeg: 21-Jun-2011
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Keel: eng
  • ISBN-13: 9783642204388
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Classical Summation in Commutative and Noncommutative Lp-SpacesSuurem pilt
  • Formaat: PDF+DRM
  • Sari: Lecture Notes in Mathematics 2021
  • Ilmumisaeg: 21-Jun-2011
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Keel: eng
  • ISBN-13: 9783642204388
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The aim of this research is to develop a systematic scheme that makes it possible to transform important parts of the by now classical theory of summation of general orthonormal series into a similar theory for series in noncommutative $L_p$-spaces constructed over a noncommutative measure space (a von Neumann algebra of operators acting on a Hilbert space together with a faithful normal state on this algebra).

This book develops a systematic scheme that makes it possible to transfer important parts of the theory of summation of general orthonormal series into a similar theory for series in noncommutative Lp-spaces built over a noncommutative measure space.

Arvustused

From the reviews:

The book under review is a beautiful and original exposition on the topic of almost everywhere convergent orthonormal series. The student or researcher who succeeds in reading this book will be rewarded with a deep understanding of the subject, both in the commutative and noncommutative setting. the book should stand on the shelf of anyone seriously interested in functional analysis and/or probability. (Stanisaw Goldstein, zbMATH, Vol. 1267, 2013)

This book is well written, with a concise, clear and readable style. It is divided into 3 chapters and includes a preface, a bibliography consisting of 98 items, and symbol, author and subject indexes. The book is a good source for specialists and graduate students working in functional analysis and operator theory. (Mohammad Sal Moslehian, Mathematical Reviews, Issue 2012 d)

1 Introduction
1(14)
1.1 Step 1
2(2)
1.2 Step 2
4(2)
1.3 Step 3
6(7)
1.3.1 Coefficient Tests in the Algebra Itself
7(2)
1.3.2 Tests in the Hilbert Spaces L2 (M, φ) and More Generally Haagerup Lp's
9(2)
1.3.3 Tests in Symmetric Spaces E (M, τ), τ a Trace
11(2)
1.4 Preliminaries
13(2)
2 Commutative Theory
15(64)
2.1 Maximizing Matrices
15(25)
2.1.1 Summation of Scalar Series
16(3)
2.1.2 Maximal Inequalities in Banach Function Spaces
19(2)
2.1.3 (p, q)-Maximizing Matrices
21(3)
2.1.4 Maximizing Matrices and Orthonormal Series
24(1)
2.1.5 Maximizing Matrices and Summation: The Case q < p
25(3)
2.1.6 Banach Operator Ideals: A Repetitorium
28(5)
2.1.7 Maximizing Matrices and Summation: The Case q ≥ p
33(4)
2.1.8 Almost Everywhere Summation
37(3)
2.2 Basic Examples of Maximizing Matrices
40(25)
2.2.1 The Sum Matrix
40(6)
2.2.2 Riesz Matrices
46(4)
2.2.3 Cesaro Matrices
50(5)
2.2.4 Kronecker Matrices
55(6)
2.2.5 Abel Matrices
61(1)
2.2.6 Schur Multipliers
62(3)
2.3 Limit Theorems in Banach Function Spaces
65(14)
2.3.1 Coefficient Tests in Banach Function Spaces
65(9)
2.3.2 Laws of Large Numbers in Banach Function Spaces
74(5)
3 Noncommutative Theory
79(80)
3.1 The Tracial Case
79(44)
3.1.1 Symmetric Spaces of Operators
80(6)
3.1.2 Maximal Inequalities in Symmetric Spaces of Operators
86(9)
3.1.3 Tracial Extensions of Maximizing Matrices
95(5)
3.1.4 The Row+Column Maximal Theorem
100(9)
3.1.5 Almost Uniform Convergence
109(2)
3.1.6 Coefficient Tests in Symmetric Operator Spaces
111(5)
3.1.7 Laws of Large Numbers in Symmetric Operator Spaces
116(2)
3.1.8 A Counterexample
118(5)
3.2 The Nontracial Case
123(36)
3.2.1 Haagerup Lp-Spaces
123(3)
3.2.2 Maximal Inequalities in Haagerup Lp-Spaces
126(2)
3.2.3 Nontracial Extensions of Maximizing Matrices
128(1)
3.2.4 Almost Sure Convergence
129(5)
3.2.5 Coefficient Tests in Haagerup Lp-Spaces
134(5)
3.2.6 Laws of Large Numbers in Haagerup Lp-Spaces
139(2)
3.2.7 Coefficient Tests in the Algebra Itself
141(7)
3.2.8 Laws of Large Numbers in the Algebra Itself
148(2)
3.2.9 Maximal Inequalities in the Algebra Itself
150(9)
References 159(6)
Symbols 165(2)
Author Index 167(2)
Subject Index 169
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