The Hermite property of a causal Wiener algebra used in control theory
type: Item
instance of: scholarly article
title: The Hermite property of a causal Wiener algebra used in control theory
zbMATH Open document ID: 1207.46044
published in: Complex Analysis and Operator Theory
publication date: 2010-10-04T00:00:00Z
review text: The paper under review considers a certain Banach algebra that arises in control theory and gives an application of the main result that is useful in stabilization of linear systems. The algebra \(\mathcal{A}\) considered consists of complex valued functions on the closed right half plane \(\{s\in \mathbb{C}: \text{Re}\,s \geq 0\}\) which are of the form \(f(s)= \widehat{f_a}(s)+ \sum_0^{\infty} f_k e^{t_k.s}\) with \(f_a \in L^1(0,\infty)\) where \(\widehat{f_a}\) denotes the Laplace transform and \((f_k)\) is a strictly increasing summable sequence with \(f_0=0\). It is a Banach algebra with pointwise operations and norm given by \(\|f\| = \|f_a\|_{L^1} + \|(f_k)\|_{\ell^1}\). The main result is that the Gelfand maximal ideal space of \(\mathcal{A}\) is contractible. The application of this mentioned in the beginning is the result that \(\mathcal{A}\) is a Hermite ring, that is, if \(f\) is a left invertible \(n\times k\) matrix over \(\mathcal{A}\), \(k<n,\) then there are \(n\times n\) matrices \(F,G\) over \(\mathcal{A}\) such that \(F_{ij}=f_{ij}\) for all \(1\leq i \leq n, 1\leq j\leq k\) and \(GF=I_n\).
Mathematics Subject Classification ID: 46J10
Mathematics Subject Classification ID: 43A20
Mathematics Subject Classification ID: 93C05
Mathematics Subject Classification ID: 46J20
zbMATH DE Number: 5795221
zbMATH Keywords: maximal ideal space
zbMATH Keywords: contractibility
zbMATH Keywords: Hermite ring
zbMATH Keywords: Callier-Desoer algebra
zbMATH Keywords: Tolokonnikov's lemma
zbMATH Keywords: coprime factorisation
zbMATH Keywords: Gel'fand maximal ideal space
author: Amol Sasane
reviewed by: K. Parthasarathy
MaRDI profile type: Publication
full work available at URL: s11785-008-0078-6
OpenAlex ID: W1967971114
DOI: 10.1007/S11785-008-0078-6
description: scientific article; zbMATH DE number 5795221