Spectral mapping theorems for subnormal operators
Let S be a subnormal operator whose minimal normal extension has scalar- valued spectral measure \(\mu\). Denote the \(weak^*\) closure in \(L^{\infty}(\mu)\) of the rational functions in z with poles off \(\sigma\) (S) by \(R^{\infty}(\sigma(S),\mu)\). For \(f\in R^{\infty}(\sigma(S),\mu)\), f(S) has a natural definition and the resulting map \(f\to f(S)\) has many pleasant properties. The paper under review investigates the problem of characterizing the various closed spectra of f(S) in terms of f and the various closed spectra of S. Given \(f\in R^{\infty}(\sigma(S),\mu)\), using function theory, a natural way is found to define the image under f of \(\sigma\) (S), denoted \(f(\sigma\) (S)), and the cluster set of f on \(\sigma_ e(S)\) denoted \(c\ell(f;\sigma_ e(S))\). Notice that for K a compact superset of \(\sigma\) (S), \(R^{\infty}(K,\mu)\subseteq R^{\infty}(\sigma(S),\mu)\) where \(R^{\infty}(K,\mu)\) denotes the obvious. Call K almost open iff the set of nonpeak points for R(K) that lie off the interior of K has area zero. Call K nice iff all but countably many points of the boundary of K are points of positive lower capacity density for the complement of K. It is proved that if K is an almost open compact superset of \(\sigma\) (S), then for any \(f\in R^{\infty}(K,\mu), f(\sigma(S))\subseteq \sigma(f(S))\) and \(c\ell(f;\sigma_ e(S))\subseteq \sigma_ e(f(S)).\) It is also proved that if K is a nice compact superset of \(\sigma\) (S), then for any \(f\in R^{\infty}(K,\mu)\), these inclusions become equalities. An example is produced to show that \(\sigma_{ap}(f(S))\) cannot be expressed solely in terms of f and \(\sigma_{ap}(S)\). Finally, some open problems are discussed. Throughout use is made of the function theory of Chaumat, Gamelin, and Garnett. \(\{\) The following two sentences were inadvertently omitted and should begin the first paragraph on page 379: ''Consequently we may suppose that \(z_ n\in \sigma_ e(S)\) for only finitely many n. Dropping these n, we may assume that \(\{z_ n\}_{n\geq 1}\subseteq E\backslash \sigma_ e(S)\subseteq \sigma(S)\backslash \sigma_ e(S).\)''\(\}\)
scientific article; zbMATH DE number 3855803