Asymptotic behaviours and generalized Toeplitz operators
The paper deals with certain generalized Toeplitz operators associated to operators \(T\) on a Hilbert space \(H\). In particular, when considering bounded linear operators \(T \in B(H)\) for which the sequence \({(\| T^n h\| )}_{n\in {\mathbb N}}\) converges for every \(h \in H\), the authors present conditions for \(T\) to be orthogonally mean ergodic. Moreover, the asymptotic limit \(S_T\) of such a \(T\) is written in terms of the ergodic projections of \(T\) and \(T^\ast\), and the unitary part for \(T\) in \(H\) is obtained. Some results of [\textit{R. G. Douglas}, Acta Sci. Math. 30, 19--32 (1969; Zbl 0177.19204)] concerning the generalized Toeplitz operators for contractions are extended. To this end, the authors apply the theory to operators of the class \(C_\rho\) (\(\rho > 0\)) in the sense of \textit{B. Sz.-Nagy} and \textit{C. Foias} [``Harmonic analysis of operators on Hilbert spaces'', Budapest: Akadémiai Kiadó, Amsterdam-London: North-Holland (1970; Zbl 0201.45003)].
scientific article; zbMATH DE number 5362153
scientific article