Classes of Linear Operators Volume 1 and 2

Preface to Volume II. Table of contents of Volume II. Introduction. PART V: TRIANGULAR REPRESENTATIONS XX. Additive lower-upper triangular decompositions of operators 1. Additive lower-upper triangular decompositions relative to finite chains 2. Preliminaries about chains 3. Diagonals 4. Chains on Hilbert space 5. Triangular algebras 6. Riemann-Stieltjes integration along chains 7. Additive lower-upper decomposition theorem 8. Additive lower-upper decomposition of a Hilbert-Schmidt operator 9. Multiplicative integration along chains 10. Basic properties of reproducing kernel Hilbert spaces and chains 11. Example of an additive LU-decomposition in a RKHS. XXI. Operators in triangular form 1. Triangular representation 2. Intermezzo about completely nonselfadjoint operators 3. Volterra operators with a one-dimensional imaginary part 4. Unicellular operators. XXII. Multiplicative lower-upper triangular decompositions of operators 1. LU-factorization with respect to a finite chain 2. The LU-factorization theorem 3. LU-factorizations of compact perturbations of the identity 4. LU-factorizatioris of Hilbert-Schmidt perturbations of the identity 5. LU-factorizations of integral operators 6. Triangular representations of operators close to unitary 7. LU-factorization of semi-separable integral operators 8. Generalised Wiener-Hopf equations 9. Generalised LU-factorization relative to discrete chains. Comments on Part V Exercises to Part V PART VI: CLASSES OF TOEPLITZ OPERATORS XXIII. Block Toeplitz operators 1 Preliminaries 2. Block Laurent operators 3. Block Toeplitz operators 4. Block Toeplitz operators defined by continuous functions 5. The Fredholm index of a block Toeplitz operator defined by a continuous function. XXIV. Toeplitz operators defined by rational matrix functions 1. Preliminaries 2. Invertibility and Fredholm index (scalar case) 3. Wiener-Hopf factorization 4. Invertibility and Fredholm index (matrix case) 5. Intermezzo about realisation 6. Inversion of a block Laurent operator 7. Explicit canonical factorization 8. Explicit inversion formulas 9. Explicit formulas for Fredholm characteristics 10. An example 11. Asymptotic formulas for determinants of block Toeplitz matrices. XXV. Toeplitz operators defined by piecewise continuous matrix functions 1. Piecewise continuous functions 2. Symbol and Fredholm index (scalar case) 3. Symbol and Fredholm index (matrix case) 4. Sums of products of Toeplitz operators defined by piecewise continuous functions 5. Sums of products of block Toeplitz operators defined by piecewise continuous functions. Comments on Part VI. Exercises to Part VI. PART VII: CONTRACTIVE OPERATORS AND CHARACTERISTIC OPERATOR FUNCTIONS XXVI. Block shift operators 1. Forward shifts and isometries 2. Parts of block shift operators 3. Invariant subspaces of forward shift operators. XXVII. Dilation theory 1. Preliminaries about contractions 2. Preliminaries about dilations 3. Isometric dilations 4. Unitary dilations