Operators with Brownian unitary dilations

Abstract

Certain bounded linear operators $T$ on a complex Hilbert space $\h$ which have 2-isometric liftings $S$ on another space $\ka \supset \h$ are being investigated. We refer also to a more special type of liftings, as well as to those which additionally meet the condition $S^*S\h \subset \h$. Furthermore we describe other types of dilations for $T$, which are close to 2-isometries. Among these we refer to expansive (concave) operators and also to Brownian unitary dilations. Different matrix representations for such operators are obtained, where matrix entries involve contractive operators.