A class of operators similar to the shift on \(H^2(G)\) (Q860214)

In his unpublished thesis [``Quasisimilarity and subnormal operators'' (Ph.\,D.\ thesis, Univ.\ of Michigan, Ann Arbor, MI (1973)], \textit{W.\,Clary} gave necessary and sufficient conditions for a cyclic subnormal operator to be quasisimilar to the unilateral unweighted shift, which is identified as the operator of multiplication by \(z\) on the classical Hardy space \(H^2(\mathbb{D})\) on the unit open disc \(\mathbb{D}\). This result has been extended by several authors to different settings [see, for instance, \textit{W.\,H.\thinspace Hastings}, Trans.\ Am.\ Math.\ Soc.\ 256, 145--161 (1979; Zbl 0381.47015); \textit{J.\,E.\thinspace McCarthy}, J.~Oper.\ Theory 24, No.\,1, 105--116 (1990; Zbl 0773.47010); \textit{J.\,Z.\thinspace Ziu}, J.~Oper.\ Theory 32, No.\,1, 47--75 (1994; Zbl 0823.47025)]. Let \(G\) be a bounded open subset whose complement in the plane has a finite number of components and for which the following conditions hold: (i) The algebra \(A(\overline{G})\) of all functions continuous on \(G\) and analytic on the interior of \(G\) coincides with the algebra \(R(\overline{G})\), the uniform closure of all rational functions with poles off \(\overline{G}\). (ii) Every analytic bounded function on \(G\) is the pointwise limit of a bounded sequence of functions in \(R(\overline{G})\). In the paper under review, the author characterizes all subnormal operators similar to \(M_z\), the operator of multiplication by \(z\) on the Hardy space \(H^2(G)\). In the case when each component of \(G\) is simply connected, he also classifies all bounded linear operators that are unitarily equivalent to \(M_z\).