This is a collection of original and review articles on recent advances and new directions in a multifaceted and interconnected area of mathematics and its applications. It encompasses many topics in theoretical developments in operator theory and its diverse applications in applied mathematics, physics, engineering, and other disciplines. The purpose is to bring in one volume many important original results of cutting edge research as well as authoritative review of recent achievements, challenges, and future directions in the area of operator theory and its applications. The intended audience are mathematicians, physicists, electrical engineers in academia and industry, researchers and graduate students, that use methods of operator theory and related fields of mathematics, such as matrix theory, functional analysis, differential and difference equations, in their work.

Many developments on the cutting edge of research in operator theory and its applications are reflected in this collection of original and review articles. Particular emphasis lies on highlighting the interplay between operator theory and applications from other areas, such as multi-dimensional systems and function theory of several complex variables, distributed parameter systems and control theory, mathematical physics, wavelets, and numerical analysis.

This volume is dedicated to Leonid Lerer on the occasion of his seventieth birthday. The main part presents recent results in Lerer's research area of interest, which includes Toeplitz, Toeplitz plus Hankel, and Wiener-Hopf operators, Bezout equations, inertia type results, matrix polynomials, and related areas in operator and matrix theory. Biographical material and Lerer's list of publications complete the volume.

This book provides a careful treatment of the theory of algebraic Riccati equations. It consists of four parts: the first part is a comprehensive account of necessary background material in matrix theory including careful accounts of recent developments involving indefinite scalar products and rational matrix functions. The second and third parts form the core of the book and concern the solutions of algebraic Riccati equations arising from continuous and discrete systems. The geometric theory and iterative analysis are both developed in detail. The last part of the book is an exciting collection of eight problem areas in which algebraic Riccati equations play a crucial role. These applications range from introductions to the classical linear quadratic regulator problems and the discrete Kalman filter to modern developments in HD*W*w control and total least squares methods.

This volume is dedicated to Leonid Lerer on the occasion of his seventieth birthday. The main part presents recent results in Lerer's research area of interest, which includes Toeplitz, Toeplitz plus Hankel, and Wiener-Hopf operators, Bezout equations, inertia type results, matrix polynomials, and related areas in operator and matrix theory. Biographical material and Lerer's list of publications complete the volume.

Many developments on the cutting edge of research in operator theory and its applications are reflected in this collection of original and review articles. Particular emphasis lies on highlighting the interplay between operator theory and applications from other areas, such as multi-dimensional systems and function theory of several complex variables, distributed parameter systems and control theory, mathematical physics, wavelets, and numerical analysis.

Thefollowing topics ofmathematical analysishavebeen developed in the last?fty years: thetheoryoflinearcanonicaldi?erentialequationswithperiodicHamilto- ans, the theory of matrix polynomials with selfadjoint coe?cients, linear di?er- tial and di?erence equations of higher order with selfadjoint constant coe?cients, andalgebraicRiccati equations.All of these theories, and others, arebased on r- atively recent results of linear algebra in spaces with an inde?nite inner product, i.e., linear algebra in which the usual positive de?nite inner product is replaced by an inde?nite one. More concisely, we call this subject inde?nite linear algebra. This book has the structureof a graduatetext in which chaptersof advanced linear algebra form the core. The development of our topics follows the lines of a usual linear algebra course. However, chapters giving comprehensive treatments of di?erential and di?erence equations, matrix polynomials and Riccati equations are interwoven as the necessary techniques are developed. The main source of material is our earlier monograph in this ?eld: Matrices and Inde?nite Scalar Products, 40]. The present book di?ers in objectives and material.Somechaptershavebeenexcluded, othershavebeenadded, andexercises have been added to all chapters. An appendix is also included. This may serve as a summary and refresher on standard results as well as a source for some less familiar material from linear algebra with a de?nite inner product. The theory developed here has become an essential part of linear algebra. This, together with the many signi?cant areas of application, and the accessible style, make this book useful for engineers, scientists and mathematicians al

Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations.

Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses or as a basis for a graduate course in linear algebra. The open problems can serve as research projects for undergraduates, topics for graduate students, or problems to be tackled by professional research mathematicians. The book is also an invaluable reference tool for researchers in fields where techniques based on quaternion analysis are used.