The interior of bounded point evaluations for rationally cyclic operators
Let \(T\) denote a linear bounded operators on a complex infinite separable Banach space \(X\). The operator \(T\) is called a multicyclic operator of order \(m\) if there exist \(m\) vectors \(y_1, y_2,\dots, y_m\in X\) such that \(X=\operatorname{span}\{T^ny_i:n\geq 0,\, i=1,\dots, m\}\) and if, for every family of \((m-1)\) vectors \(x_1, x_2,\dots, x_{m-1}\) in \(X\), the closed linear subspace \(\operatorname{span}\{T^nx_i:n\geq 0,\, i=1,\dots, m-1\}\) is proper. The \(m\)-tuple \((y_1, y_2, \dots, y_m)\) is then called a cyclic \(m\)-tuple for \(T\). Recall that, for \(m = 1\), \(T\) is a cyclic operator. The operator \(T\) is called a rationally \(m\)-cyclic operator if there exist \(m\) vectors \(y_1, y_2,\dots, y_m\in X\) such that \(X=\operatorname{span}\{\sum_{i=1}^m r_i(T)y_i:r_i\in {\mathcal R}(\sigma(T)),\,i=1,\dots, m\}\) and if, for every family of \((m-1)\) vectors \(x_1, x_2,\dots, x_{m-1}\) in \(X\), the closed linear subspace \(X=\operatorname{span}\{\sum_{i=1}^{m-1} r_i(T)x_i:r_i\in {\mathcal R}(\sigma(T)),\,i=1,\dots, m-1\}\) is proper. Here, \({\mathcal R}(\sigma(T))\) denotes the set of all rational functions with poles of the spectrum \(\sigma(T)\) of \(T\). Let \((y_1, y_2,\dots, y_m)\) be a cyclic \(m\)-tuple of a multicyclic operator \(T\). A complex number \(\lambda_0\in\sigma(T)\) is said to be a bounded point evaluation for \(T\) if there exists \(M>0\) such that \(\sum_{i=1}^m|P_i(\lambda_0)|\leq M\|\sum_{i=1}^m P_i(T)y_i\|\) for every family of complex polynomials \(\{P_1,P_2,\dots, P_m\}\). The set of all bounded point evaluations for \(T\) is denoted by \(B(T)\). In a similar way, \(\lambda_0\in \sigma(T)\) is a bounded point evaluation for a rationally \(m\)-cyclic operator \(T\) if \(\sum_{i=1}^m|r_i(\lambda_0)|\leq M\|\sum_{i=1}^m r_i(T)y_i\|\) for every family of rational functions \(\{r_1,r_2,\dots, r_m\}\) with poles of \(\sigma(T)\). If \(\lambda_0\in B(T)\) and \(j \in \{1,\dots, m\}\), the linear map \(w^j(\lambda_0):\{\sum_{i=1}^m r_i(T)y_i:r_i\in {\mathcal R}(\sigma(T)),\ i=1,\dots, m\}\to {\mathbb C}\), defined by \(\sum_{i=1}^m r_i(T)y_i\to r_j(\lambda_0)\), is well defined and can be extended to a bounded linear functional defined on \(X\) because \(\{\sum_{i=1}^m r_i(T)y_i:r_i\in {\mathcal R}(\sigma(T)),\,i=1,\dots, m\}\) is a dense subset of \(X\). Denote by \(k_j(\lambda_0)\) such an extension. A point \(\lambda_0\in \operatorname{int}(B(T))\) is called an analytic bounded point evaluation for \(T\) if the mapping \(\lambda\mapsto \hat{y}_i(\lambda) =\langle y,\ k_i(\lambda) \rangle \) is analytic at \(\lambda_0\) for every \(y\in X\), and for \(i = 1, 2,\dots,m\). The set of all analytic bounded point evaluations for \(T\) is denoted by \(B_a(T)\). \textit{J. B. Conway} [The theory of subnormal operators. Providence, RI: American Mathematical Society (1991; Zbl 0743.47012), 7.11, p.\,65)] asked if the interiors of \(B(T)\) and \(B_a(T)\) coincide for arbitrary subnormal operators on Hilbert spaces. In the paper under review, the authors provide an example that answers negatively Conway's question in the more general setting of operators satisfying Bishop's property \((\beta)\). In particular, they show that \(B_a(T)\setminus\Lambda = \operatorname{int}(B(T))\setminus\Lambda\), with different subsets \(\Lambda\) in \(\sigma(T)\).
scientific article; zbMATH DE number 6620737
scientific article