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Rational dilation on an annulus. (English) Zbl 0609.47013
Let \({\mathcal H}\) be a Hilbert space and \({\mathcal L}({\mathcal H})\) denote the set of continuous liner operators on \({\mathcal H}\). Let K be a compact subset of the complex plane \({\mathbb{C}}\) and let \(\text{Rat}(K)\) denote the set of rational functions with poles off \(K\). If \(T\in {\mathcal L}({\mathcal H})\) and the spectrum \(\sigma(T)\subseteq K\), we say that \(K\) is a spectral set for \(T\) if \(\| f(T)\| \leq \max \{| f(z)|:z\in K\}\) when \(f\in \text{Rat}(K)\). B. Sz-Nagy’s [Acta Sci. Math. 15, 87-92 (1953; Zbl 0052.12203)] famous dilation theorem led to the following open problem:
Let \(T\in {\mathcal L}({\mathcal H})\) and \(K\) be a compact subset of \({\mathbb{C}}\). Are the following two conditions equivalent?
(i) \(K\) is a spectral set for \(T\).
(ii) There exists a Hilbert space \({\mathcal K}\supseteq {\mathcal H}\) and a normal operator \(N\in {\mathcal L}({\mathcal H})\) such that \(\sigma(N)\subseteq \partial K\) and, if P denotes the orthogonal projection of \({\mathcal K}\) onto \({\mathcal H}\), then \(f(T)=Pf(N)|_{{\mathcal H}}\) for all \(f\in \text{Rat}(K)\).
Though this problem is still open for a general compact set \(K\), the author solves in this highly interesting paper the above problem for an annulus \(K\) given by \(K=\{z\in {\mathbb{C}}:r\leq z\leq 1,0<r<1\}\).
Reviewer: N.K.Thakare

47A20 Dilations, extensions, compressions of linear operators
47A45 Canonical models for contractions and nonselfadjoint linear operators
47A25 Spectral sets of linear operators
47A60 Functional calculus for linear operators
Zbl 0052.12203
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