Recent Developments in Operator Theory and Its Applications

The present volume contains the proceedings of the International Conference on Ap­ plications of Operator Theory held in Winnipeg, Canada (October 2nd to 6th, 1994), which was organized by the Institute of Industrial Mathematical Sciences (IIMS) of the University of Manitoba. At this conference 92 participants representing 15 countries par­ ticipated, and 64 papers were presented. This meeting was the second of a linked pair. The first was a program of advanced instruction held at the Fields Institute, Ontario, followed by a research conference. The first of these events gave rise to the volume "Lectures on Operator Theory and its Applications", published by the American Mathematical Society for the Fields Institute in 1995. These two events were the creation of the following Program Committee: M. A. Dahleh (M. I. T. ) P. A. Fillmore (Dalhousie) B. A. Francis (Toronto) F. Ghahramani (Manitoba) K. Glover (Cambridge) I. Gohberg (Tel Aviv) T. Kailath (Stanford) P. Lancaster (Calgary), Chair H. Langer (Vienna) P. N. Shivakumar (Manitoba) A. A. Shkalikov (Moscow) B. Simon (Cal. Tech. ) H. Widom (Santa Cruz) Both events focused on the following main topics: Infinite matrices and projection methods, linear operators on indefinite scalar product spaces, differential operators and mathematical systems theory and control. This volume contains a selection of papers in modern operator theory and its appli­ cations. They are dedicated to recent achievements and many are written by leaders in the mentioned fields.

Inverse scattering problem for continuous transmission lines with rational reflection coefficient function.- 1. Introduction.- 2. Some results on differential expressions.- 3. The reflection coefficient function.- 4. The rational case.- References.- The band method and the Grassmannian approach for completion and extension problems.- 1. Introduction.- 2. The Grassmannian approach.- 3. Proofs of theorems 2.2 and 2.3.- 4. The Caratheodory and Nehari extension problems.- 5. Operator matrix extension problems.- References.- Polar decompositions in finite dimensional indefinite scalar product spaces: Special cases and applications.- 1. Introduction.- 2. H-contractive matrices.- 3. H-plus matrices.- 4. Indefinite scalar products with only one positive square.- 5. Polar decompositions with special unitary factors.- 6. Applications: Linear optics.- References.- Positive differential operators in Krein space L2(R).- 1. Abstract results.- 2. Differential operators with constant coefficients.- 3. Half-range completeness.- References.- Elliptic problems involving an indefinite weight function.- 1. Introduction.- 2. Preliminaries.- 3. Main results.- 4. Examples.- References.- The KDV hierarchy and associated trace formulas.- 1. Introduction.- 2. The KDV hierarchy, recursion relations, and hyperelliptic curves.- 3. The stationary formalism.- 4. The time-dependent formalism.- 5. General trace formulas.- Appendix A. Hyperelliptic curves of the KDV-type and theta functions.- Appendix B. An explicit illustration of the Riemann-Roch theorem.- References.- On spectral properties of Schrödinger-type operator with complex potential.- 1. General perturbation results.- 2. Application to the Schrodinger-type operator.- Appendix 1. Proof of Theorem 1.- Appendix 2. Proof of Theorem 6.- Appendix 3. The subordination condition.- References.- Coalgebras and spectral theory in one and several parameters.- 1. Introduction.- 2. Coalgebras and comodules.- 3. The coalgebra dual of a polynomial algebra.- 4. The primary decomposition theorem.- 5. Monic matrix polynomials.- 6. Several commuting maps.- 7. Multiparameter systems.- References.- Destabilization of infinite-dimensional time-varying systems via dynamical output feedback.- 1. Introduction.- 2. Notation and fundamental results.- 3. System description.- 4. Problem formulation.- 5. Destabilization results.- References.- Perturbations of G-selfadjoint operators and operator polynomials with real spectrum.- 1. Introduction.- 2. Perturbations of finite rank.- 3. Small and compact perturbations.- 4. Applications to operator polynomials.- 5. A factorization theorem in the monic case.- 6. Differential equations with stably bounded solutions.- 7. The case of noninvertible leading coefficient.- References.- Definitizable G-unitary operators and their applications to operator polynomials.- 1. Introduction.- 2. Preliminary definitions and results.- 3. Compact perturbations.- 4. Operator polynomials quasihyperbolic on T.- 5. Other charcterizations of QHP on T.- References.- System theoretic aspects of completely symmetric systems.- 1. Introduction.- 2. Discrete time systems.- 3. Continuous-time systems.- References.- Contractive completion of block matrices and its application to H? control of periodic systems.- 1. Introduction.- 2. H? periodic control and lifting.- 3. Matrix contractive completion.- 4. All H? suboptimal periodic controllers.- 5. Concluding remarks.- Appendix: Proof of Theorem 1.- References.- Spline approximation methods for Wiener-Hopf operators.- 1. Introduction.- 2. Technical preliminaries.- 3. An algebra of approximation sequences for Wiener-Hopf operators.- 4. Approximation methods for composed operators.- References.- Inertia conditions for the minimization of quadratic forms in indefinite metric spaces.- 1. Introduction.- 2. An inertia result for linear transformations.- 3. The indefinite-weighted least-squares problem.- 4. The equivalent estimation problem.- 5. Relations between the IWLS and EE problems.- 6. Incorporating state-space structure.- 7. A recursive IWLS problem in the presence of state-space structure.- 8. An application to H?-filtering.- 9. An application to robust adaptive filters.- 10. An application to total least-squares methods.- 11. Concluding remarks.- References.- Bounds for the width of the instability intervals in the Mathieu equation.- 1. Introduction.- 2. Preliminaries.- References.- Operator pencils arising in elasticity and hydrodynamics: The instability index formula.- 1. Classes of unbounded operator pencils.- 2. Root subspaces of linear dissipative pencils and their properties.- 3. Quadratic dissipative pencils and the instability index formula.- 4. Applications.- References.- Toeplitz-like operators and their finite sections.- 1. Introduction.- 2. The main results.- 3. Proofs.- References.- Spectral representations and spectral functions of symmetric operators.- 0. Introduction.- 1. Spectral representations of linear operators.- 2. Selfadjoint extension in a larger Hilbert space.- 3. The generalized spectral function of a symmetric operator and the corresponding spectral transformation.- 4. Generalized resolvents of a symmetric operator.- References.- Hankel type operators, Bourgain algebras, and isometries.- 1. Hankel type operators and Bourgain algebras.- 2. Complete continuity of Hankel type operators and isometries.- 3. Biholomorphic equivalence and Bourgain algebras.- References.- Effective computation of operators defined by line integrals.- 1. Introduction.- 2. Numerical procedure.- 3. Modified Gaussian quadratures.- 4. Piecewise modified Guassian quadrature on an interval.- 5. The calculation of line integrals.- 6. Extension for the calculation of operators defined by Cauchy principal value integrals.- References.