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Quaternionic de Branges spaces and c...

Alpay, Daniel.

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Quaternionic de Branges spaces and characteristic operator function

Record Type:	Electronic resources : Monograph/item
Title/Author:	Quaternionic de Branges spaces and characteristic operator function/ by Daniel Alpay, Fabrizio Colombo, Irene Sabadini.
Author:	Alpay, Daniel.
other author:	Colombo, Fabrizio.
Published:	Cham :Springer International Publishing : : 2020.,
Description:	x, 116 p. :ill., digital ;24 cm.
[NT 15003449]:	Preliminaries -- Quaternions and matrices -- Slice hyperholomorphic functions -- Rational functions -- Operator models -- Structure theorems for H(A;B) spaces -- J-contractive functions -- The characteristic operator function -- L() spaces and linear fractional transformations -- Canonical differential systems.
Contained By:	Springer eBooks
Subject:	Functions, Quaternion. -
Online resource:	https://doi.org/10.1007/978-3-030-38312-1
ISBN:	9783030383121

Quaternionic de Branges spaces and characteristic operator function
Alpay, Daniel.

Quaternionic de Branges spaces and characteristic operator function[electronic resource] /by Daniel Alpay, Fabrizio Colombo, Irene Sabadini. - Cham :Springer International Publishing :2020. - x, 116 p. :ill., digital ;24 cm. - SpringerBriefs in mathematics,2191-8198. - SpringerBriefs in mathematics..

Preliminaries -- Quaternions and matrices -- Slice hyperholomorphic functions -- Rational functions -- Operator models -- Structure theorems for H(A;B) spaces -- J-contractive functions -- The characteristic operator function -- L() spaces and linear fractional transformations -- Canonical differential systems.

This work contributes to the study of quaternionic linear operators. This study is a generalization of the complex case, but the noncommutative setting of quaternions shows several interesting new features, see e.g. the so-called S-spectrum and S-resolvent operators. In this work, we study de Branges spaces, namely the quaternionic counterparts of spaces of analytic functions (in a suitable sense) with some specific reproducing kernels, in the unit ball of quaternions or in the half space of quaternions with positive real parts. The spaces under consideration will be Hilbert or Pontryagin or Krein spaces. These spaces are closely related to operator models that are also discussed. The focus of this book is the notion of characteristic operator function of a bounded linear operator A with finite real part, and we address several questions like the study of J-contractive functions, where J is self-adjoint and unitary, and we also treat the inverse problem, namely to characterize which J-contractive functions are characteristic operator functions of an operator. In particular, we prove the counterpart of Potapov's factorization theorem in this framework. Besides other topics, we consider canonical differential equations in the setting of slice hyperholomorphic functions and we define the lossless inverse scattering problem. We also consider the inverse scattering problem associated with canonical differential equations. These equations provide a convenient unifying framework to discuss a number of questions pertaining, for example, to inverse scattering, non-linear partial differential equations and are studied in the last section of this book.

ISBN: 9783030383121

Standard No.: 10.1007/978-3-030-38312-1doiSubjects--Topical Terms:

2058363
Functions, Quaternion.

LC Class. No.: QA196 / .A473 2020

Dewey Class. No.: 512.5

Quaternionic de Branges spaces and characteristic operator function
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100 1 $a Alpay, Daniel. $3 713947
245 1 0 $a Quaternionic de Branges spaces and characteristic operator function $h [electronic resource] / $c by Daniel Alpay, Fabrizio Colombo, Irene Sabadini.
260 $a Cham : $b Springer International Publishing : $b Imprint: Springer, $c 2020.
300 $a x, 116 p. : $b ill., digital ; $c 24 cm.
490 1 $a SpringerBriefs in mathematics, $x 2191-8198
505 0 $a Preliminaries -- Quaternions and matrices -- Slice hyperholomorphic functions -- Rational functions -- Operator models -- Structure theorems for H(A;B) spaces -- J-contractive functions -- The characteristic operator function -- L() spaces and linear fractional transformations -- Canonical differential systems.
520 $a This work contributes to the study of quaternionic linear operators. This study is a generalization of the complex case, but the noncommutative setting of quaternions shows several interesting new features, see e.g. the so-called S-spectrum and S-resolvent operators. In this work, we study de Branges spaces, namely the quaternionic counterparts of spaces of analytic functions (in a suitable sense) with some specific reproducing kernels, in the unit ball of quaternions or in the half space of quaternions with positive real parts. The spaces under consideration will be Hilbert or Pontryagin or Krein spaces. These spaces are closely related to operator models that are also discussed. The focus of this book is the notion of characteristic operator function of a bounded linear operator A with finite real part, and we address several questions like the study of J-contractive functions, where J is self-adjoint and unitary, and we also treat the inverse problem, namely to characterize which J-contractive functions are characteristic operator functions of an operator. In particular, we prove the counterpart of Potapov's factorization theorem in this framework. Besides other topics, we consider canonical differential equations in the setting of slice hyperholomorphic functions and we define the lossless inverse scattering problem. We also consider the inverse scattering problem associated with canonical differential equations. These equations provide a convenient unifying framework to discuss a number of questions pertaining, for example, to inverse scattering, non-linear partial differential equations and are studied in the last section of this book.
650 0 $a Functions, Quaternion. $3 2058363
650 1 4 $a Functional Analysis. $3 893943
650 2 4 $a Operator Theory. $3 897311
650 2 4 $a Functions of a Complex Variable. $3 897310
700 1 $a Colombo, Fabrizio. $3 1536005
700 1 $a Sabadini, Irene. $3 927744
710 2 $a SpringerLink (Online service) $3 836513
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830 0 $a SpringerBriefs in mathematics. $3 1566700
856 4 0 $u https://doi.org/10.1007/978-3-030-38312-1
950 $a Mathematics and Statistics (Springer-11649)