Opuscula Mathematica

Abstract

The sub-Brownian 3-isometries in Hilbert spaces are the natural counterparts of the 2-isometries, because all of them have Brownian-type extensions in the sense of J. Agler and M. Stankus. We show that all powers $T^n$ for $n\geq 2$ of every expansive 3-isometry $T$ are sub-Brownian, even if $T$ does not have such a property. This fact induces some useful relations between the corresponding covariance operators of $T$. We analyze two matrix representations of $T$ in order to get some conditions under which $T$ is sub-Brownian, or $T$ admits the Wold-type decomposition in the sense of S. Shimorin. We show that the restriction of $T$ to its range is sub-Brownian of McCullough's type, and that under some conditions on $\mathcal{N}(T^*)$, $T$ itself is sub-Brownian, and it admits the Wold-type decomposition.Krakówwersja wydawnicz