The interior of bounded point evaluations for rationally cyclic operators
type: Item
instance of: scholarly article
title: The interior of bounded point evaluations for rationally cyclic operators
zbMATH Open document ID: 1448.47017
author: Mostafa Mbekhta
author: Nadia Ourchane
author: E. H. Zerouali
published in: Mediterranean Journal of Mathematics
publication date: 2016-08-31T00:00:00Z
review text: 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):\{\su…
Mathematics Subject Classification ID: 47A16
Mathematics Subject Classification ID: 47B37
Mathematics Subject Classification ID: 47A13
Mathematics Subject Classification ID: 47B20
zbMATH DE Number: 6620737
zbMATH Keywords: rationally \(m\)-cyclic operators
zbMATH Keywords: analytic bounded point evaluations
zbMATH Keywords: analytic core
MaRDI profile type: Publication
full work available at URL: s00009-015-0585-4
OpenAlex ID: W977859904
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DOI: 10.1007/S00009-015-0585-4
description: scientific article; zbMATH DE number 6620737