The Hermite property of a causal Wiener algebra used in control theory
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 \(GF=I_n\).
scientific article; zbMATH DE number 5795221