On the Nehari problem for a certain class of \(L^{\infty}\) functions appearing in control theory. II
type: Item
instance of: scholarly article
title: On the Nehari problem for a certain class of \(L^{\infty}\) functions appearing in control theory. II
zbMATH Open document ID: 0686.47004
DOI: 10.1016/0022-1236(88)90098-5
author: Allen Tannenbaum
author: Ciprian Foias
published in: Journal of Functional Analysis
publication date: 1988-01-01T00:00:00Z
review text: In the previous paper of the authors [J. Funkt. Anal. 74, 146-159 (1987; Zbl 0635.47027)] an explicit formula was given for the computation of the norm of the operator w(T) where w is a rational function belonging to \(H^{\infty}\) and T denotes the compression of the unilateral shift on \(H^ 2\) to \(H^ 2\theta mH^ 2\) for an inner function m in \(H^{\infty}\) \((H^ p:=H^ p(D)\) for the unit disc D). It turns out that this formula is a special case of a formula valid for T an arbitrary contraction on a Hilbert space. Indeed, an explicit procedure is given to compute the isolated spectrum of \(W=w(T)w(T)^*\). If \(T\in C_ 0\), a relatively simple expression is derived even for the discrete spectrum of W. The results apply to control engineering problems. The main tool is furnished by Theorem 1: Suppose \(w=p/q\) where p, q are relatively prime polynomials, none of which is a Blaschke product, and set \(n=\max \{\deg p\), deg \(q\}\). Let \(R={\mathbb{R}}^+\setminus \{spectrum\) of \(W\}\), and for \(r\in R\) set \(P_ r=q(T)(1-r^{-2}W)q(T)^*\). Further, let v(r) be the spectral radius of \((I-T^ nT^{*^ n})P_ r^{-1}\) It is known that v(r) equals \(\| D_{T^*}P_ r^{- 1}D_{T^*}\|\) where \(D_ A=(I-A^*A)^{1/2}\). The main new result is that if \(\bar r\) is a point on the boundary of R different of the points (if any) of the points of the spectrum of T lying on the unit circle, then \(v(z)\to +\infty\) for \(z\in R\), \(x\to \bar r\).
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47A45
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 46J15
zbMATH DE Number: 4124411
zbMATH Keywords: contraction operators on Hilbert space
zbMATH Keywords: Hardy classes
zbMATH Keywords: spectral theory
zbMATH Keywords: computation of the norm
zbMATH Keywords: unilateral shift
zbMATH Keywords: Blaschke product
MaRDI profile type: Publication
full work available at URL: 0022-1236(88)90098-5
OpenAlex ID: W2047383999
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description: scientific article; zbMATH DE number 4124411