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 applications of traces and determi­ nants of infinite matrices and integral operators. We were especially impressed by the papers of Poincare, von Koch, Fredholm, Hilbert and Carleman, as well as F. Riesz's book on infinite systems of linear equations. Consequently our plans were changed and we decided to write a paper on the history of determinants of infi­ nite matrices and operators. During the preparation of our paper we realized that many mathematical questions had to be answered in order to gain a more com­ plete understanding of the subject. So, we changed our plans again and decided to present the subject in a more advanced form which would satisfy our new require­ ments. This whole process took between four and five years of challenging, but enjoyable work. This entailed the study of the appropriate relatively recent results of Grothendieck, Ruston, Pietsch, Hermann Konig and others. After the papers [GGK1] and [GGK2] were published, we saw that the written material could serve as the basis of a book.

Table of Contents:
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 withcontinuous 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 operatorswith 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.