Weak closure of the unitary orbit of contractions
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\(\to L(H)\); \(\psi (f)=f(T)\).
scientific article; zbMATH DE number 3953629