# Journal of Operator Theory

Volume 53, Issue 1, Winter 2005 pp. 49-89.

Generalized Toeplitz operators, restrictions to invariant subspaces and similarity problems**Authors**: Gilles Cassier

**Author institution:**Institut Girard Desargues, UMR 5028 du CNRS, UFR de Math\'ematiques, B\^at. Jean Braconnier, Université Claude Barnard Lyon I, F-69622 Villeurbanne Cedex, France

**Summary:**Our purpose is to investigate the asymptotic properties of an operator $T$ on an invariant subspace $E\in \mathrm{Lat}(T)$ and on $E^\perp $ using the generalized Toeplitz operators associated with $T$. We show how the relative properties may be used in order to give a general result linking the behaviour of $T$ on $E$ and on $E^\perp $ with the possibility for $T$ to be similar to a scalar multiple of a contraction. Some applications are indicated. In particular, one of our results implies that there is no hope to construct a power bounded operator of Foguel type that is not similar to a contraction and such that for every $x\in H\backslash \{0\}$ the sequence $% (T^{n})_{n\geqslant 0}$ does not converge to $0$. We also study the asymptotic and spectral properties of these operators of Foguel type.

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