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.- 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 $${\mathcal{D}_\Omega }(\mathcal{H})$$. Hill's 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 Hill's 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 algebr

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