The authors prove that for a linear contraction T on a Hilbert space H there exists an isometry Q on H and another linear contraction R on H such that \(\lim_{n\to \infty}\| T^ nf-Q^ nRf\| =0\) for all \(f\in H\). They give an example T of a linear contraction on all \(L_ p(\mu)\)-spaces, \(1\leq p\infty <\infty\), such that \(T1=1\) with the property: \(Tf>0\) for all \(f\geq 0\), \(f\neq 0\) for which the set \[ \{f\in L_ p(\mu):\quad \| T^ nf\|_ p=\| f\|_ p\quad for\quad all\quad n\geq 0\} \] contains only constant functions for any \(1<p<\infty\) but there is an \(f\in L_{\infty}(\mu)\) such that \(T^ nf\nrightarrow \mu (f)\) in \(L_ 1(\mu)\)-norm. This example is related to an example given by M. Rosenblatt [Markov processes, Structure and asymptotic behavior (1971; Zbl 0236.60002), pp. 115-115].