Asymptotic behaviours and generalized Toeplitz operators

We study some generalized Toeplitz operators associated to operators T on a Hilbert space H, for which there exists the limit of {‖Tnh‖} for every h∈H. We refer to the asymptotic limit ST of such a T, in the sense of [L. Kerchy, Operators with regular norm-sequences, Acta Sci. Math. (Szeged) 63 (1997) 571–605; L. Kerchy, Generalized Toeplitz operators, Acta Sci. Math. (Szeged) 68 (2002) 373–400; G. Cassier, Generalized Toeplitz operators, restrictions to invariant subspaces and similarity problems, J. Operator Theory 53 (1) (2005) 101–140; C.S. Kubrusly, An Introduction to Models and Decompositions in Operator Theory, Birkhäuser, Boston, 1997], and we give some conditions of ergodicity for T. Also, certain results of Douglas [R.G. Douglas, On the operator equation S∗XT=X and related topics, Acta Sci. Math. (Szeged) 30 (1969) 19–32] involving generalized Toeplitz operators are extended in our more general setting, and we apply these results to ρ-contractions.