Ergodic properties for regular \(A\)-contractions (Q850896)

Let \(H\) be a complex Hilbert space and \(B(H)\) be the Banach algebra of all its bounded linear operators. An operator \(T\) in \(B(H)\) is called an \(A\)-contraction for a positive operator \(A\) in \(B(H)\) if \(T^* AT \leq A\); an \(A\)-contraction is called regular if \(AT = A^{1/2}TA^{1/2}\). In this paper, an orthogonal decomposition of \(H\) induced by a regular \(A\)-contraction is obtained and this is used to prove a certain version of the mean ergodic theorem and Patil's theorem. Some functional equations in the range of \(A^{1/2}\) are characterized analogously to the result of [\textit{M.\,Lin} and \textit{R.\,Sine}, J.~Oper.\ Theory 10, 153--166 (1983; Zbl 0553.47006)].