Weak closure of the unitary orbit of contractions
type: Item
instance of: scholarly article
title: Weak closure of the unitary orbit of contractions
zbMATH Open document ID: 0593.47032
DOI: 10.1007/BF01955737
published in: Acta Mathematica Hungarica
publication date: 1985-01-01T00:00:00Z
review text: Let H be an infinite dimensional complex separable Hilbert space. For a bounded linear operator T(T\(\in L(H))\) on H; U(T), W(T), \(\sigma\) (T), \(\sigma_ e(T)\), WC(U(T)) and \((HB)_ 1\) denote, unitary orbit, numerical range, spectrum, essential spectrum, of T, weak closure of U(T), and set of all contractions on H. The following theorems are proved. Theorem 1. For any contraction T on H \((\| T\| \leq 1)\), the following are equivalent a) \(WCU(T)=(HB)_ 1,b)\) \(\overline{W(T)}=\bar D,\bar D=closed\) unit disc, c) \(\sigma\) (T)\(\supset \partial \bar D\), \(\partial \bar D=boundary\) of \(\bar D,\) d) \(\sigma_ e(T)\supset \partial \bar D.\) This theorem generalizes a result proved by \textit{P. R. Halmos}, Acta Sci. Math. 34, 131-139 (1973; Zbl 0257.47019)]. Theorem 2. For any contractive weighted shift \(T_{\alpha}\) with positive weights \((\alpha_ i)\) on H; \(WCU(T_{\alpha})=(HB)_ 1\) if, and only if, \(\forall n\epsilon {\mathbb{N}}\), \(\epsilon >0\), \(\exists N\epsilon {\mathbb{N}}\) such that \(\alpha_{N+i}>1-\epsilon\), \(i=1,2,...,n.\) The paper contains some more interesting results for arbitrary weighted shifts. Moreover it contains an application to the disc algebra A namely; for \(T\epsilon\) L(H), \(\| T\| \leq 1\), \(WC(U(T))=(HB)_ 1\) if, and only if \(\psi\) is an isometry, where \(\psi\) :A\(\to L(H)\); \(\psi (f)=f(T)\).
Mathematics Subject Classification ID: 47B37
Mathematics Subject Classification ID: 47A12
Mathematics Subject Classification ID: 47A25
zbMATH DE Number: 3953629
zbMATH Keywords: unitary orbit
zbMATH Keywords: numerical range
zbMATH Keywords: essential spectrum
zbMATH Keywords: set of all contractions
zbMATH Keywords: contractive weighted shift
author: Mahmoud Kutkut
MaRDI profile type: Publication
cites work: Q5660202
cites work: Q5673462
cites work: Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes. Erhard Schmidt zum 75. Geburtstag in Verehrung gewidmet
description: scientific article; zbMATH DE number 3953629