This volume presents the proceedings of the Joint German-Israeli Workshop on linear one-dimensional singular integral equations, held in Tel Aviv from March 1–10, 1995. The volume contains a selection of papers in modern operator theory and its applications. The main topics of the workshop were symbol calculus, index formulas, projection and quadrature methods for Toeplitz and singular integral operators with different symbols, algebras generated by such operators and algebras generated by indempotents. The other topics discussed were inverse scattering problems for differential operators, distribution of zeros for orthogonal functions, factorization of matrix functions and calculation of norms. The book will be appreciated by a wide audience in the mathematical and engineering sciences.
Inverse scattering problem for differential operators with rational scattering matrix functions.- 1. Introduction.- 2. Rational matrix valued functions.- 3. Minimal realizations of rational scattering functions.- 4. Marchenko’s equation in the rational case.- 5. Formula for the potential.- 6. Equivalence with Krein’s approach.- 7. A result on asymptotic expansions.- 8. Reconstructing the potential.- References.- Banach algebras generated by N idempotents and applications.- Preface and acknowledgement.- 1. Introduction.- 2. Algebras generated by two idempotents.- 3. Algebras generated by three idempotents.- 4. An N projections theorem.- 5. Examples.- 6. Miscellanea.- References.- Toeplitz operators with discontinuous symbols: Phenomena beyond piecewise continuity.- 1. Introduction.- 2. Preliminaries.- 3. Periodic and u-periodic symbols.- 4. Orientation preserving whirls.- 5. Concrete orientation preserving whirls.- 6. Orientation changing whirls.- 7. Undecided symbols with bounded arguments.- 8. Modulated almost periodic symbols.- References.- Toeplitz and singular integral operators on general Carleson Jordan curves.- 1. Introduction.- 2. The first three metamorphoses.- 3. Main result.- 4. Examples.- 5. Plan of the proof of the main theorem.- 6. Submultiplicative functions.- 7. Description of the set Nt.- 8. Description of the set ?t.- 9. Spectra and essential spectra of Toeplitz operators.- 10.The algebra of singular integral operators.- References.- Weighted uniform convergence of the quadrature method for Cauchy singular integral equations.- 1. Introduction.- 2. Notations and preliminary results.- 3. The quadrature method.- 4. Special interpolation processes.- 5. Numerical examples.- References.- Symbol calculus for singular integrals with operator-valued PQC-coefficients.- 1. Introduction.- 2. Operator-valued quasicontinuous functions.- 3. Symbol calculus.- References.- Finite section method for singular integrals with operator-valued PQC-coefficients.- 1. Introduction.- 2. Preliminaries.- 3. A generalization of compactness and of strong convergence.- 4. Algebraization of stability and lifting theorem.- 5. The algebras S(PQCL(H)) and F(PQCL(H)).- 6. Localization.- 7. Identification of local algebras.- 8. The algebras ? and ?.- 9. Identification of local algebras continued.- 10.Final result.- References.- Distribution of zeros of orthogonal functions related to the Nehari problem.- 1. Introduction.- 2. Auxilliary theorems.- 3. Proof of the main theorem.- References.- Semi-Fredholm properties of certain singular integral operators.- 1. Introduction.- 2. Operators with almost periodic symbols.- 3. Operators with semi almost periodic symbols.- References.- On canonical factorization of dissipative and positive matrix functions relative to non-simple contours.- References.- Asymptotic invertibility of Toeplitz operators.- 1. Introduction.- 2. Operators.- 3. Finite sections and algebras.- 4. Symbols.- 5. Pseudospectra.- 6. Asymptotic Moore-Penrose invertibility.- References.- Calculation of the norm of polynomials of two adjoint projections.- 0. Introduction.- 1. Some necessary results.- 2. The main results.- 3. Additional remarks and examples.- References.- Toeplitz Lectures 1995.- Lectures of the Joint German-Israeli Workshop.
