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Operator Methods for Boundary Value Problems
Presented in this volume are a number of new results concerning the extension theory and spectral theory of unbounded operators using the recent notions of boundary triplets and boundary relations. This approach relies on linear single-valued and multi-valued maps, isometric in a Krein space sense, and offers a basic framework for recent developments in system theory. Central to the theory are analytic tools such as Weyl functions, including Titchmarsh-Weyl m-functions and Dirichlet-to-Neumann maps. A wide range of topics is considered in this context from the abstract to the applied, including boundary value problems for ordinary and partial differential equations; infinite-dimensional perturbations; local point-interactions; boundary and passive control state/signal systems; extension theory of accretive, sectorial and symmetric operators; and Calkin's abstract boundary conditions. This accessible treatment of recent developments, written by leading researchers, will appeal to a broad range of researchers, students and professionals.
Regular Boundary Value Problems Associated with Pairs of Ordinary Differential Expressions
It is well known that two hermitian n x n matrices K, H, where H is positive definite, H> 0, can be simultaneously diagonalized. The key to the proof is to consider C[superscript]n, where C is the complex number field, as a Hilbert space [Fraktur capital]H [subscript]H with the inner product given by (f, g) = g*Hf, where f, g [lowercase Greek]Epsilon C[superscript]n, considered as a space of column vectors. Then the operator A = H−1K is selfadjoint in [Fraktur capital]H [subscript]H, and the spectral theorem readily yields the result. Of course such A, when K is not hermitian, can also be investigated in [Fraktur capital]H [subscript]H. We consider a similar problem where K, H are replaced by a pair of ordinary differential expressions L and M, where M> 0 in some sense. Two difficulties arise: (1) there are many natural choices for a selfadjoint H> 0 generated by M, and hence many choices for [Fraktur capital]H [subscript]H, and (2), once a choice for H has been made, there are many choices for the analogue of A. In our work we consider all possible choices for H> 0 and the analogue of A.
Multivariate Prediction, de Branges Spaces, and Related Extension and Inverse Problems
This monograph deals primarily with the prediction of vector valued stochastic processes that are either weakly stationary, or have weakly stationary increments, from finite segments of their past. The main focus is on the analytic counterpart of these problems, which amounts to computing projections onto subspaces of a Hilbert space of p x 1 vector valued functions with an inner product that is defined in terms of the p x p matrix valued spectral density of the process. The strategy is to identify these subspaces as vector valued de Branges spaces and then to express projections in terms of the reproducing kernels of these spaces and/or in terms of a generalized Fourier transform that is ob...
Holomorphic Spaces
Expository articles describing the role Hardy spaces, Bergman spaces, Dirichlet spaces, and Hankel and Toeplitz operators play in modern analysis.
Schur Functions, Operator Colligations, and Reproducing Kernel Pontryagin Spaces
Generalized Schur functions are scalar- or operator-valued holomorphic functions such that certain associated kernels have a finite number of negative squares. This book develops the realization theory of such functions as characteristic functions of coisometric, isometric, and unitary colligations whose state spaces are reproducing kernel Pontryagin spaces. This provides a modern system theory setting for the relationship between invariant subspaces and factorization, operator models, Krein-Langer factorizations, and other topics. The book is intended for students and researchers in mathematics and engineering. An introductory chapter supplies background material, including reproducing kernel Pontryagin spaces, complementary spaces in the sense of de Branges, and a key result on defining operators as closures of linear relations. The presentation is self-contained and streamlined so that the indefinite case is handled completely parallel to the definite case.
Mathematical Reviews
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Trends in Theory and Practice of Nonlinear Differential Equations
This book is based on an International Conference on Trends in Theory and Practice of Nonlinear Differential Equations held at The University of Texas at Arlington. It aims to feature recent trends in theory and practice of nonlinear differential equations.
