On the Nehari problem for a certain class of \(L^{\infty}\) functions appearing in control theory. II
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 \(z\in R\), \(x\to \bar r\).
scientific article; zbMATH DE number 4124411