Journal article
Rational Cayley inner Herglotz-Agler functions: Positive-kernel decompositions and transfer-function realizations
Linear algebra and its applications, v 456
01 Sep 2014
Featured in Collection : UN Sustainable Development Goals @ Drexel
Abstract
The Bessmertnyi class consists of rational matrix-valued functions of d complex variables representable as the Schur complement of a block of a linear pencil A(z) = z(1)A(1) + ... + z(d)A(d) whose coefficients A(k) are positive semidefinite matrices. We show that it coincides with the subclass of rational functions in the Herglotz-Agler class over the right poly-halfplane which are homogeneous of degree one and which are Cayley inner. The latter means that such a function is holomorphic on the right poly-halfplane and takes skew-Hermitian matrix values on (iR)(d), or equivalently, is the double Cayley transform (over the variables and over the matrix values) of an inner function on the unit polydisk. Using Agler-Knese's characterization of rational inner Schur-Agler functions on the polydisk, extended now to the matrix-valued case, and applying appropriate Cayley transformations, we obtain characterizations of matrix-valued rational Cayley inner Herglotz-Agler functions both in the setting of the polydisk and of the right poly-halfplane, in terms of transfer-function realizations and in terms of positive-kernel decompositions. In particular, we extend Bessmertnyrs representation to rational Cayley inner Herglotz-Agler functions on the right poly-halfplane, where a linear pencil A(z) is now in the form A(z) = A(0) + z(1)A(1) + ... + z(d)A(d) with A(0) skew-Hermitian and the other coefficients Ak positive semidefinite matrices. (C) 2014 Elsevier Inc. All rights reserved.
Metrics
Details
- Title
- Rational Cayley inner Herglotz-Agler functions: Positive-kernel decompositions and transfer-function realizations
- Creators
- Joseph A. Ball - Virginia TechDmitry S. Kaliuzhnyi-Verboyetskyi - Drexel Univ, Dept Math, Philadelphia, PA 19104 USA
- Publication Details
- Linear algebra and its applications, v 456
- Publisher
- Elsevier
- Number of pages
- 19
- Grant note
- 2010432 / US-Israel BSF; US-Israel Binational Science Foundation DMS-0901628 / NSF; National Science Foundation (NSF)
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000338808400011
- Scopus ID
- 2-s2.0-84903270717
- Other Identifier
- 991019168293204721
UN Sustainable Development Goals (SDGs)
This publication has contributed to the advancement of the following goals:
InCites Highlights
Data related to this publication, from InCites Benchmarking & Analytics tool:
- Collaboration types
- Domestic collaboration
- Web of Science research areas
- Mathematics
- Mathematics, Applied