Logo image
Back
Schur–Agler and Herglotz–Agler classes of functions: Positive-kernel decompositions and transfer-function realizations
Journal article   Open access   Peer reviewed

Schur–Agler and Herglotz–Agler classes of functions: Positive-kernel decompositions and transfer-function realizations

Joseph A. Ball and Dmitry S. Kaliuzhnyi-Verbovetskyi
Advances in mathematics (New York. 1965), v 280, pp 121-187
06 Aug 2015
DOI: https://doi.org/10.1016/j.aim.2015.04.018
Featured in Collection :   UN Sustainable Development Goals @ Drexel
url
https://doi.org/10.1016/j.aim.2015.04.018View
Published, Version of Record (VoR)Open Access (Publisher-Specific),  Open

Abstract

Bessmertnyĭ long resolvent representation Herglotz–Agler class Lurking-isometry method Lurking-isotropic-subspace method Positive-kernel decomposition Schur–Agler class Transfer-function realization
We discuss transfer-function realization for multivariable holomorphic functions mapping the unit polydisk or the right polyhalfplane into the operator analogue of either the unit disk or the right halfplane (Schur/Herglotz functions over either the unit polydisk or the right polyhalfplane) which satisfy the appropriate stronger contractive/positive real part condition for the values of these functions on commutative tuples of strict contractions/strictly accretive operators (Schur–Agler/Herglotz–Agler functions over either the unit polydisk or the right polyhalfplane). As originally shown by Agler, the first case (polydisk to disk) can be solved via unitary extensions of a partially defined isometry constructed in a canonical way from a kernel decomposition for the function (the lurking-isometry method). We show how a geometric reformulation of the lurking-isometry method (embedding of a given isotropic subspace of a Kreĭn space into a Lagrangian subspace—the lurking-isotropic-subspace method) can be used to handle the second two cases (polydisk to halfplane and polyhalfplane to disk), as well as the last case (polyhalfplane to halfplane) if an additional growth condition at ∞ is imposed. For the general fourth case, we show how a linear-fractional-transformation change of variable can be used to arrive at the appropriate symmetrized nonhomogeneous Bessmertnyĭ long-resolvent realization. We also indicate how this last result recovers the classical integral representation formula for scalar-valued holomorphic functions mapping the right halfplane into itself.

Metrics

11 Record Views
16 citations in Web of Science
17 citations in Scopus

Details

UN Sustainable Development Goals (SDGs)

This publication has contributed to the advancement of the following goals:

#11 Sustainable Cities and Communities

InCites Highlights

Data related to this publication, from InCites Benchmarking & Analytics tool:

Collaboration types
Domestic collaboration
Web of Science research areas
Mathematics
Logo image