Abstract. There exists a unitary operator $V$ in $L_2[0, 1)$ such that, for any increasing sequence of indices $n(k)$, the averages ${1\over K} \sum_{1\leq k\leq K}V^{n(k)}f$ do not converge a.s. for some function $f\in L_2[0, 1).$

AMS Subject Classification (1991): 47A35

Keyword(s): subsequence, ergodic means, ergodic type theorem

Received January 19, 2004, and in revised form August 2, 2005. (Registered under 5897/2009.)