Counterexamples to rational dilation on symmetric multiply connected domains (Q1958654)

For \(X\) a compact, path connected subset of \(\mathbb{C}\), we denote by \(\mathcal{R}(X)\) the space of all rational functions with poles off \(X\). For a bounded operator \(T\in\mathcal{B}(H)\) on a Hilbert space \(H\), we say that \(T\) has a normal \(\partial X\)-dilation if there is a bounded normal operator \(N\) on a Hilbert space \(K\supseteq H\) such that \(\sigma(N)\subseteq\partial X\) and \(f(T)=P_{H}f(N)|_{H}\) for all \(f\in\mathcal{R}(X)\). If \(T\) has a normal \(\partial X\)-dilation, then \(X\) is a spectral set for \(T\), that is, \(\sigma(T)\subseteq X\) and \(\|f(T)\|\leq\|f\|_{X}=\sup\{|f(z)|:z\in X\}\) for all \(f\in\mathcal{R}(X)\). The converse of this is known as the \textit{rational dilation conjecture}. Sz.-Nagy's well-known dilation theorem shows that the rational dilation conjecture holds if \(X\) is the closed unit disc [\textit{B.\,Sz-Nagy}, Acta Sci.\ Math.\ 15, 87--92 (1953; Zbl 0052.12203)]. A generalization by \textit{C.\,Berger} [Ph.\ D.\ thesis, Cornell Univ.\ (1963)], \textit{C.\,Foiaş} [Studii Cerc.\ Mat.\ 10, 365--401 (1959; Zbl 0091.27903)] and \textit{A.\,Lebow} [J.~Math.\ Anal.\ Appl.\ 7, 64--90 (1963; Zbl 0145.39301)] shows that this holds for any simply connected domain in the plane. \textit{J.\,Agler} [Ann.\ Math.\ (2) 121, 537--563 (1985; Zbl 0609.47013)] showed that rational dilation also holds if \(X\) has one hole, such as for an annulus. The aim of the paper under review is to prove the following, which by a corollary to Arveson's extension theorem, is equivalent to showing that the rational dilation conjecture does not hold on symmetric domains with two or more holes. Main Theorem. Let \(X\subseteq\mathbb{C}\) be a symmetric, two-or-more-holed domain with analytic boundary consisting of disjoint curves. Then there is a bounded operator \(T\) on some Hilbert space \(H\) such that the homomorphism \(\pi:\mathcal{R}(X)\rightarrow\mathcal{B}(H)\) defined by \(\pi(p/q)=p(T)q(T)^{-1}\) is contractive but not completely contractive.