On decomposition for pairs of twisted contractions

TL;DR

This paper develops Wold-type decompositions for pairs of twisted contractions on Hilbert spaces, providing new proofs, explicit decompositions, and characterizations for various classes of twisted operators and their pairs.

Contribution

It introduces novel Wold-type decompositions for pairs of twisted contractions, including doubly twisted isometries, and offers explicit decompositions and characterizations, along with simplified proofs of existing theorems.

Findings

Wold-type decomposition for pairs of twisted contractions

Explicit decomposition for pairs with c.n.u. parts in C_{00}

Doubly twisted isometries characterized and related to pairs of isometries

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 S\l{}o\'{c}inski'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 $(T,V^*)$ 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.