Author | Gohberg, Israel. author |
---|---|
Title | Traces and Determinants of Linear Operators [electronic resource] / by Israel Gohberg, Seymour Goldberg, Nahum Krupnik |
Imprint | Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 2000 |
Connect to | http://dx.doi.org/10.1007/978-3-0348-8401-3 |
Descript | IX, 258 p. online resource |
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 Poincarรฉ determinant -- 2 Hillโs method -- 3 The Von Koch-Riesz algebra -- 4 The Mennicken-Wagenfรผhrer algebra -- 5 The algebra D(?1) -- 6 Comments -- 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 with continuous and piecewise continuous kernels -- 2 The Algebra $${\mathcal{D}_\Omega }(\mathcal{H})$$. Hillโs Method (revisited) -- 3 Diagonally modified Fredholm determinant -- 4 A modification of the Plemelj-Smithies formula -- 5 Integral Operators in L1(T, ?, ?) -- 6 Systems of integral equations -- 7 Comments -- VII Possible Values of Traces and Determinants. Perelson Algebras -- 1 Perelson algebras -- 2 Possible values of traces and determinants in Perelson algebras -- 3 Possible values in $${\mathcal{D}_\Omega }(\mathcal{H})$$ -- 4 Comments -- VIII Inversion Formulas -- 1 General inversion formulas -- 2 Explicit formulas for resolvents of integral operators -- 3 Homogeneous integral equations -- 4 Comments -- IX Regularized Determinants -- 1 Extension problems -- 2 The main (extension) theorems for regularized determinants -- 3 Analyticity, Plemelj-Smithies formulas -- 4 Comments -- X Hilbert-Carleman Determinants -- 1 Integral operators with degenerate kernels -- 2 Integral operators on a class of Banach spaces -- 3 Hilbert-Schmidt integral operators -- 4 Mikhlin-Itskovich algebra -- 5 Algebra ?1 -- 6 Diagonally modified Hilbert-Carleman determinant -- 7 Hilbert-Carleman determinant for infinite matrices -- 8 Comments -- XI Regularized Determinants of Higher Order -- 1 Main extension theorems -- 2 Analyticity and Plemelj-Smithies formulas -- 3 Preparation for the proof of Theorem IV.10.3 -- 4 Proof of Theorem IV.10.3 -- 5 Comments -- XII Inversion Formulas via Generalized Determinants -- 1 General case -- 2 Integral equations -- 3 Systems of Hillโs equations -- 4 Comments -- XIII Determinants of Integral Operators with Semi-separable Kernels -- 1 Statement of the main theorem -- 2 Input-output representations -- 3 Cascade connection of systems -- 4 Inverse systems -- 5 Inversion of integral operators with semi-separable kernels -- 6 Indicator of integral operators -- 7 Computation of the Hilbert-Carleman and the Fredholm determinants -- 8 Spectra of integral operators with semi-separable kernels -- 9 Time invariant systems -- 10 Counting negative eigenvalues of a Hilbert-Schmidt operator via sign changes of a determinant -- 11 Comments -- XIV Algebras without the Approximation Property -- 1 A general class of algebras -- 2 Integral operators with a jump discontinuity on the diagonal -- 3 Applications to integral operators with a jump discontinuity on the diagonal -- 4 Applications to integral operators with semi-separable kernel -- 5 Comments -- List of Symbols