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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

MSC:
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
Citations:
Zbl 0052.12203
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