Bøker av Israel Gohberg

The tangential trigonometric moment problem on an interval and related topics.- 1. Introduction.- 2. Some lemmas on matrix-valued rational functions.- 3. The main result.- 4. The Nevanlinna-Pick problem.- References.- Maximum entropy and joint norm bounds for operator extensions.- 1. Introduction.- 2. A sharp bound in the 2×2 case.- 3. The maximum entropy method.- 4. An application to integral operators.- References.- Bitangential interpolation for input-output operators of time varying systems: the discrete time case.- 0. Introduction.- 1. Residue calculus and generalized point evaluation.- 2. Pairs of diagonal operators and homogeneous one-sided interpolation.- 3. Bitangential interpolation data set.- 4. Bitangential interpolation in geometric terms.- 5. Intermezzo about admissible Sylvester data sets.- 6. Construction of a particular solution.- 7. Parametrization of all solutions (without norm constraints).- 8. Input-output operators of time-varying systems.- 9. Parametrization of all contractive input-output operators satisfying the bitangential interpolation conditions.- References.- Two-sided tangential interpolation of real rational matrix functions.- 1. Introduction.- 2. Minimal realizations.- 3. Local data.- 4. Two-sided tangential interpolation: existence of real interpolants.- 5. Two-sided tangential interpolation with real-valued data: Description of interpolants.- 6. Degrees of interpolants.- 7. Generalized Nevanlinna-Pick interpolation for real rational matrix functions.- References.- On the spectra of operator completion problems.- 1. Introduction.- 2. Case of finite dimensional spaces.- 3. Case of infinite dimensional spaces.- References.- The exact H2 estimate for the central H? interpolant.- 1. An improved Kaftal-Larson-Weiss estimate.- 2. Some formulas for DB?.- 3. The role of DA?2II0*.- 4. The four block problem.- 5. Optimal solutions.- References.- On mixed H2 - H? tangential interpolation.- 1. Introduction.- 2. Formulas for the central solution.- 3. A state space approach.- 4. Applications of the H2 - H? tangential interpolation problem.- References.- On a completion problem for matrices.- 1. Introduction.- 2. Main theorems in the finite dimensional case.- 3. The full range case.- 4. The proof of the main theorems in the finite dimensional case.- 5. Infinite dimensional case.- References.

V: Triangular Representations.- XX. Additive lower-upper triangular decompositions of operators.- XXI. Operators in triangular form.- XXII. Multiplicative lower-upper triangular decompositions of operators.- VI: Classes of Toeplitz Operators.- XXIII. Block Toeplitz operators.- XXIV. Toeplitz operators defined by rational matrix functions.- XXV. Toeplitz operators defined by piecewise continuous matrix functions.- VII: Contractive Operators and Characteristic Operator Functions.- XXVI. Block shift operators.- XXVII. Dilation theory.- XXVIII. Unitary systems and characteristic operator functions.- VIII: Banach Algebras And Algebras Of Operators.- XXIX. General theory.- XXX. Commutative Banach algebras.- XXXI. Elements of C*-algebra theory.- XXXII. Banach algebras generated by Toeplitz operators.- IX: Extension and Completion Problems.- XXXIII. Completions of matrices.- XXXIV. A general scheme for completion and extension problems.- XXXV. Applications of the band method.- Standard references texts.- List of symbols.

Uncertainty principles for time-frequency operators.- 1. Introduction.- 2. Sampling results for time-frequency transformations.- 3. Uncertainty principles for exact Gabor and wavelet frames.- References.- Distribution of zeros of matrix-valued continuous analogues of orthogonal polynomials.- 1. Preliminary results.- 1.1. Matrix-valued Krein functions of the first and second kinds.- 1.2. Partitioned integral operators.- 2. Orthogonal operator-valued polynomials.- 2.1. Stein equations for operators.- 2.2. Zeros of orthogonal polynomials.- 2.3. On Toeplitz matrices with operator entries.- 3. Zeros of mat rix-valued Krein functions.- 3.1 On Wiener-Hopf operators.- 3.2. Proof of the main theorem.- References.- The band extension of the real line as a limit of discrete band extensions, II. The entropy principle.- 0. Introduction.- I. Preliminaries.- II. Main results.- References.- Weakly positive matrix measures, generalized Toeplitz forms, and their applications to Hankel and Hilbert transform operators.- 1. Lifting properties of generalized Toeplitz forms and weakly positive matrix measures.- 2. The GBT and the theorems of Helson-Szegö and Nehari.- 3. GNS construction, Wold decomposition and abstract lifting theorems.- 4. Multiparameter and n-conditional lifting theorems, the A-A-K theorem and applications in several variables.- References.- Reduction of the abstract four block problem to a Nehari problem.- 0. Introduction.- 1. Main theorems.- 2. Proofs of the main theorems.- References.- The state space method for integro-differential equations of Wiener-Hopf type with rational matrix symbols.- 1. Introduction and main theorems.- 2. Preliminaries on matrix pencils.- 3. Singular differential equations on the full-line.- 4. Singular differential equations on the half-line.- 5. Preliminaries on realizations.- 6. Proof of theorem 1.1.- 7. Proofs of theorems 1.2 and 1.3.- 8. An example.- References.- Symbols and asymptotic expansions.- 0. Introduction.- I. Smooth symbols on Rn.- II. Piecewise smooth symbols on T.- III. Piecewise smooth symbols on Rn.- IV. Symbols discontinuous across a hyperplane in Rn × Rn.- References.- Program of Workshop.

6 Preliminaries.- 6.1 The operator of singular integration.- 6.2 The space Lp(?, ?).- 6.3 Singular integral operators.- 6.4 The spaces $$L_{p}^{ + }(\Gamma, \rho ), L_{p}^{ - }(\Gamma, \rho ) and \mathop{{L_{p}^{ - }}}\limits^{^\circ } (\Gamma, \rho )$$.- 6.5 Factorization.- 6.6 One-sided invertibility of singular integral operators.- 6.7 Fredholm operators.- 6.8 The local principle for singular integral operators.- 6.9 The interpolation theorem.- 7 General theorems.- 7.1 Change of the curve.- 7.2 The quotient norm of singular integral operators.- 7.3 The principle of separation of singularities.- 7.4 A necessary condition.- 7.5 Theorems on kernel and cokernel of singular integral operators.- 7.6 Two theorems on connections between singular integral operators.- 7.7 Index cancellation and approximative inversion of singular integral operators.- 7.8 Exercises.- Comments and references.- 8 The generalized factorization of bounded measurable functions and its applications.- 8.1 Sketch of the problem.- 8.2 Functions admitting a generalized factorization with respect to a curve in Lp(?, ?).- 8.3 Factorization in the spaces Lp(?, ?).- 8.4 Application of the factorization to the inversion of singular integral operators.- 8.5 Exercises.- Comments and references.- 9 Singular integral operators with piecewise continuous coefficients and their applications.- 9.1 Non-singular functions and their index.- 9.2 Criteria for the generalized factorizability of power functions.- 9.3 The inversion of singular integral operators on a closed curve.- 9.4 Composed curves.- 9.5 Singular integral operators with continuous coefficients on a composed curve.- 9.6 The case of the real axis.- 9.7 Another method of inversion.- 9.8 Singular integral operators with regel functions coefficients.- 9.9 Estimates for the norms of the operators P?, Q? and S?.- 9.10 Singular operators on spaces H?o(?, ?).- 9.11 Singular operators on symmetric spaces.- 9.12 Fredholm conditions in the case of arbitrary weights.- 9.13 Technical lemmas.- 9.14 Toeplitz and paired operators with piecewise continuous coefficients on the spaces lp and ?p.- 9.15 Some applications.- 9.16 Exercises.- Comments and references.- 10 Singular integral operators on non-simple curves.- 10.1 Technical lemmas.- 10.2 A preliminary theorem.- 10.3 The main theorem.- 10.4 Exercises.- Comments and references.- 11 Singular integral operators with coefficients having discontinuities of almost periodic type.- 11.1 Almost periodic functions and their factorization.- 11.2 Lemmas on functions with discontinuities of almost periodic type.- 11.3 The main theorem.- 11.4 Operators with continuous coefficients - the degenerate case.- 11.5 Exercises.- Comments and references.- 12 Singular integral operators with bounded measurable coefficients.- 12.1 Singular operators with measurable coefficients in the space L2(?).- 12.2 Necessary conditions in the space L2(?).- 12.3 Lemmas.- 12.4 Singular operators with coefficients in ?p(?). Sufficient conditions.- 12.5 The Helson-Szegö theorem and its generalization.- 12.6 On the necessity of the condition a ? Sp.- 12.7 Extension of the class of coefficients.- 12.8 Exercises.- Comments and references.- 13 Exact constants in theorems on the boundedness of singular operators.- 13.1 Norm and quotient norm of the operator of singular integration.- 13.2 A second proof of Theorem 4.1 of Chapter 12.- 13.3 Norm and quotient norm of the operator S? on weighted spaces.- 13.4 Conditions for Fredholmness in spaces Lp(?, ?).- 13.5 Norms and quotient norm of the operator aI + bS?.- 13.6 Exercises.- Comments and references.- References.

My Life and Mathematics.- List of Publications of Peter Lancaster.- Forty-four Years with Peter Lancaster.- Peter Lancaster, my Friend and Co-author.- The Joint Numerical Range of Bordered and Tridiagonal Matrices.- Iterative Computation of Higher Derivatives of Repeated Eigenvalues and the Corresponding Eigenvectors.- Colligations in Pontryagin Spaces with a Symmetric Characteristic Function.- Logarithmic Residues of Fredholm Operator Valued Functions and Sums of Finite Rank Projections.- Positive Linear Maps and the Lyapunov Equation.- Full-and Partial-Range Completeness.- Spectral Isomorphisms between Generalized Sturm-Liouville Problems.- Existence and Uniqueness Results for Nonlinear Cooperative Systems.- Young's Inequality in Compact Operators.- Partial Indices of Small Perturbations of a Degenerate Continuous Matrix Function.- Finite Section Method for Difference Equations.- Iterative Solution of a Matrix Riccati Equation Arising in Stochastic Control.- Lyapunov Functions and Solutions of the Lyapunov Matrix Equation for Marginally Stable Systems.- Invariant Subspaces of Infinite Dimensional Hamiltonians and Solutions of the Corresponding Riccati Equations.- Inertia Bounds for Operator Polynomials.- A Note on the Level Sets of a Matrix Polynomial and its Numerical Range.

Nevanlinna-Pick interpolation for time-varying input-output maps: The discrete case.- 0. Introduction.- 1. Preliminaries.- 2. J-Unitary operators on ?2.- 3. Time-varying Nevanlinna-Pick interpolation.- 4. Solution of the time-varying tangential Nevanlinna-Pick interpolation problem.- 5. An illustrative example.- References.- Nevanlinna-Pick interpolation for time-varying input-output maps: The continuous time case.- 0. Introduction.- 1. Generalized point evaluation.- 2. Bounded input-output maps.- 3. Residue calculus and diagonal expansion.- 4. J-unitary and J-inner operators.- 5. Time-varying Nevanlinna-Pick interpolation.- 6. An example.- References.- Dichotomy of systems and invertibility of linear ordinary differential operators.- 1. Introduction.- 2. Preliminaries.- 3. Invertibility of differential operators on the real line.- 4. Relations between operators on the full line and half line.- 5. Fredholm properties of differential operators on a half line.- 6. Fredholm properties of differential operators on a full line.- 7. Exponentially dichotomous operators.- 8. References.- Inertia theorems for block weighted shifts and applications.- 1. Introduction.- 2. One sided block weighted shifts.- 3. Dichotomies for left systems and two sided systems.- 4. Two sided block weighted shifts.- 5. Asymptotic inertia.- 6. References.- Interpolation for upper triangular operators.- 1. Introduction.- 2. Preliminaries.- 3. Colligations & characteristic functions.- 4. Towards interpolation.- 5. Explicit formulas for ?.- 6. Admissibility and more on general interpolation.- 7. Nevanlinna-Pick Interpolation.- 8. Carathéodory-Fejér interpolation.- 9. Mixed interpolation problems.- 10. Examples.- 11. Block Toeplitz & some implications.- 12. Varying coordinate spaces.- 13. References.- Minimality and realization of discrete time-varying systems.- 1. Preliminaries.- 2. Observability and reachability.- 3. Minimality for time-varying systems.- 4. Proofs of the minimality theorems.- 5. Realizations of infinite lower triangular matrices.- 6. The class of systems with constant state space dimension.- 7. Minimality and realization for periodical systems.- References.