Abstract. A result of Eckstein asserts that for any $\rho $-contraction $T$ on a Hilbert space $\mathcal{H}$ the sequence $(||T^{n}h||)_{n}$ is convergent for any $h\in\mathcal {H}$. We show that this remains true for a natural generalization of the class of $\rho $-contractions, which we call the class of $(\rho,N)$-contractions (notation: $\mathcal{C}_{\rho,N}(\mathcal{H})$). Our argument follows the lines of Mlak's proof of Eckstein's result, but is somewhat simplified by a study of coisometric $(\rho,N)$-dilations of these operators, which seems to be of independent interest. Along the way we also point out that Gavruta's example extends to the class of $(\rho,N)$-contractions. Namely, let $\mathcal{C}_{\infty,\infty }(\mathcal{H}) :=\cup_{\rho,N}\mathcal{C}_{\rho,N}(\mathcal{H})$; then, for any integer $p>1$, there exists an operator $T$ such that $T^{p}=I$ and $T\notin\mathcal {C}_{\infty,\infty }(\mathcal{H})$.

DOI: 10.14232/actasm-014-004-2

AMS Subject Classification (1991): 47A20

Keyword(s): contractions, coisometric dilations, similarity

Received July 18, 2014, and in final form January 5, 2015. (Registered under 4/2014.)