Classes of operators related to isometries (Q1989146)

Let \(H\) be a complex Hilbert space and \(B(H)\) be the algebra of all bounded linear operators on \(H\). The authors define certain extensions of isometric operators. An operator \(T\in B(H)\) is said to be an analytic extension of an isometric operator if there exists an invariant subspace \(M\) such that \(T\vert_{M} \) is isometric and the restriction of \(T^{*} \) on \(M^{\bot} \) is algebraic. It is shown that, if \(T\) is an analytic extension of an isometric operator, then \(T\) is subscalar of order \(2n+2\); also, \(T\) satisfies Bishop's property (\(\beta\)), satisfies the single-valued extension property (SVEP), and is polaroid. Similar results are proved for operators which are \(F\)-quasi isometric. These are the operators \(T\) for which \(F(T)^{*} (T^{*} T-I)F(T)=0\) for some non-constant analytic function \(F\) defined on a neighbourhood of the spectrum \(\sigma (T)\).