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This book is the second edition of the first complete study and monograph dedicated to singular traces. The text offers, due to the contributions of Albrecht Pietsch and Nigel Kalton, a complete theory of traces and their spectral properties on ideals of compact operators on a separable Hilbert space. The second edition has been updated on the fundamental approach provided by Albrecht Pietsch. For mathematical physicists and other users of Connes' noncommutative geometry the text offers a complete reference to traces on weak trace class operators, including Dixmier traces and associated formulas involving residues of spectral zeta functions and asymptotics of partition functions.
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"Each chapter begins with a summary of the main results and ends with comments and historical notes, which makes the reading pleasant despite the sometimes heavy technical details." Mathematical Reviews
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Content
- Intro
- Preface
- Notations
- Contents
- Introduction
- Part I: Preliminary material
- 1 What is a singular trace?
- 2 Singular values and submajorization
- Part II: Theory of traces on ideals of L ( H)
- Introduction
- 3 Calkin correspondence for norms and traces
- 4 Pietsch correspondence
- 5 Spectrality of traces
- Part III: Formulas for traces on L⊂&1,8&/sub&
- Introduction
- 6 Dixmier traces and positive traces
- 7 Diagonal formulas for traces
- 8 Heat trace and ?-function formulas
- 9 Criteria for measurability
- A Miscellaneous results
- Bibliography
- Index
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