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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
Zbl 0052.12203
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