On decomposition for pairs of twisted contractions

Abstract

This paper presents Wold-type decomposition for various pairs of twisted contractions on Hilbert spaces. As a consequence, we obtain Wold-type decomposition for pairs of doubly twisted isometries and in particular, new and simple proof of Słoćinski's theorem for pairs of doubly commuting isometries are provided. We also achieve an explicit decomposition for pairs of twisted contractions such that the c.n.u. parts of the contractions are in $C_{00}$. It is shown that for a pair $(T,V^{\ast })$ of twisted operators with $T$ as a contraction and $V$ as an isometry, there exists a unique (upto unitary equivalence) pair of doubly twisted isometries on the minimal isometric dilation space of $T$. As an application, we prove that pairs of twisted operators consisting of an isometry and a co-isometry are doubly twisted. Finally, we have given a characterization for pairs of doubly twisted isometries.

arXiv comment29 pages. Revised and section 6 is added