Traces and Determinants of Linear Operators
The authors initially planned to write an article describing the origins and devel opments of the theory of Fredholm operators and to present their recollections of this topic. We started to read again classical papers and we were sidetracked by the literature concerned with the theory and applicat...
| Main Authors: | , , |
|---|---|
| Format: | eBook |
| Language: | English |
| Published: |
Basel
Birkhäuser
2000, 2000
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| Edition: | 1st ed. 2000 |
| Series: | Operator Theory: Advances and Applications
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- IV Trace Class and Hilbert-Schmidt Operators in Hilbert Space
- 1 Preliminaries
- 2 Singular numbers
- 3 Inequalities for eigenvalues, diagonal elements and singular numbers
- 4 Additional inequalities for singular numbers
- 5 Ideal of trace class operators
- 6 Lidskii trace theorem
- 7 Hilbert-Schmidt operators
- 8 Tests of nuclearity for integral operators with continuous and Hilbert-Schmidt kernels
- 9 Integral operators with smooth kernels
- 10 Polynomial operator pencils
- 11 Classes Sp
- 12 Comments
- V Nuclear Operators in Banach Spaces
- 1 The Ruston-Grothendieck algebra of nuclear operators
- 2 Examples of nuclear operators in Banach spaces
- 3 Grothendieck trace theorem
- 4 Asymptotic behavior of eigenvalues of nuclear operators
- 5 Comments
- VI The Fredholm Determinant
- 1 The Fredholm determinant for integral operators withcontinuous and piecewise continuous kernels
- 2 The Algebra
- I Finite Rank Operators
- 1 Trace and determinant for finite rank operators
- 2 Properties of the trace and determinant
- 3 Representations of the trace and determinant
- 4 Uniqueness of the trace and determinant
- 5 Von Koch form of the determinant
- 6 Fredholm form of the determinant
- 7 Plemelj-Smithies formulas
- 8 Polynomial operator pencils
- 9 Inversion formulas
- 10 Comments
- II Continuous Extension of Trace and Determinant
- 1 Extension problems and embedded algebras
- 2 Main theorems
- 3 Analyticity of the determinant and the Plemelj-Smithies formulas
- 4 Lipschitz conditions
- 5 Several remarks
- 6 Connections between the zeros of the determinant and the eigenvalues of an operator
- 7 Determinants of infinite matrices in Von Koch form
- 8 Comments
- III First Examples
- 1 The Poincaré determinant
- 2 Hill’s method
- 3 The Von Koch-Riesz algebra
- 4 The Mennicken-Wagenführer algebra
- 5 The algebra D(?1)
- 6 Comments