Traces and determinants of linear operators线性操作的迹和行列式

Preface
Introduction．
Ⅰ Fillite 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 V0n Koch form of the determinant．
　6 Fredholnl form of the determinant．
　7 Plemelj-Smithies formulas
　8 Polynomial operator pencils．
　9 Inversion formulas
　10 CommentS
ⅡContinuous 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 Poincar6 determinant
　2 Hill’S method．
　3 The Von Koch—Riesz algebra
　4 The Mennicken-Wagenffthrer algebra
　5 The algebra*
　6 Comments
Ⅳ Trace Class and Hilbert-Schmldt 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 Predholm determinant for integral operators with Continuous and piecewise continuous kernels
　2 The Algebra *Hill’S Method(revisited)．
　3 Diagonally modified Predholm determinant
　4 A modification of the Plemelj—Smithies formula
　5 Integral Operators in La（T，∑，u）
　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 in D．
　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
Ⅹ Hilbert-Carleman Determinants
Ⅺ Regularized Determinants of Higher Order
Ⅻ Inversion Formulas via Generalised Determinants
ⅩⅢ Determinants of Integral Operators with Semi-separable Kernels
ⅩⅣ Algebras without the Approximation Property
Bibliography
Index
List of Symbols