Operator Theory: Advances and Applications

6 primary works

Schur Functions, Operator Colligations and Reproducing Kernel Pontryagin Spaces

by Daniel Alpay

Published June 1997

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 reproduced kernal 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. This text is intended for students and researchers in mathematics and engineering. An introductory chapter contains background material, including reproducing kernal Pontryagin spaces, complementary spaces in the sense of de Branges, and a result on defining operators as closures of linear relations. The presentation is such that the indefinite case is handled parallel to the definite case.

Wavelets, Multiscale Systems and Hypercomplex Analysis

by Daniel Alpay

Published 1 January 2006

This volume contains a selection of papers on the topics of Clifford analysis and wavelets and multiscale analysis, the latter being understood in a very wide sense. The theory of wavelets is mathematically rich and has many practical applications. Most of the articles have been written on invitation and they provide a unique collection of material, particularly relating to Clifford analysis and the theory of wavelets.

System Theory, the Schur Algorithm and Multidimensional Analysis

by Daniel Alpay

Published 1 January 2007

This volume contains six peer-refereed articles written on the occasion of the workshop Operator theory, system theory and scattering theory: multidimensional generalizations and related topics, held at the Department of Mathematics of the Ben-Gurion University of the Negev in June, 2005. The book will interest a wide audience of pure and applied mathematicians, electrical engineers and theoretical physicists.

Slice Hyperholomorphic Schur Analysis

by Daniel Alpay, Fabrizio Colombo, and Irene Sabadini

Published 16 December 2016

This book defines and examines the counterpart of Schur functions and Schur analysis in the slice hyperholomorphic setting. It is organized into three parts: the first introduces readers to classical Schur analysis, while the second offers background material on quaternions, slice hyperholomorphic functions, and quaternionic functional analysis. The third part represents the core of the book and explores quaternionic Schur analysis and its various applications. The book includes previously unpublished results and provides the basis for new directions of research.

Spectral Theory on the S-Spectrum for Quaternionic Operators

by Fabrizio Colombo, Jonathan Gantner, and David P. Kimsey

Published 18 January 2019

The subject of this monograph is the quaternionic spectral theory based on the notion of S-spectrum. With the purpose of giving a systematic and self-contained treatment of this theory that has been developed in the last decade, the book features topics like the S-functional calculus, the F-functional calculus, the quaternionic spectral theorem, spectral integration and spectral operators in the quaternionic setting. These topics are based on the notion of S-spectrum of a quaternionic linear operator. Further developments of this theory lead to applications in fractional diffusion and evolution problems that will be covered in a separate monograph.

Quaternionic Closed Operators, Fractional Powers and Fractional Diffusion Processes

by Fabrizio Colombo and Jonathan Gantner

Published 25 July 2019

This book presents a new theory for evolution operators and a new method for defining fractional powers of vector operators. This new approach allows to define new classes of fractional diffusion and evolution problems.

These innovative methods and techniques, based on the concept of S-spectrum, can inspire researchers from various areas of operator theory and PDEs to explore new research directions in their fields.

This monograph is the natural continuation of the book: Spectral Theory on the S-Spectrum for Quaternionic Operators by Fabrizio Colombo, Jonathan Gantner, and David P. Kimsey (Operator Theory: Advances and Applications, Vol. 270).