On decomposition for pairs of contractions

Satyabrata Majee, Amit Maji

Published 2022-08-05Version 1

This paper presents Wold-type decomposition for various pairs of commuting contractions on Hilbert spaces. As a consequence, we obtain a new and simple proof of S\l{}o\'{c}inski's theorem for pairs of doubly commuting isometries. We also achieve an explicit decomposition for pairs of commuting contractions such that the c.n.u. parts of the contractions are in $C_{00}$. It is also shown that if a pair $(T, V)$ of commuting operators with $T$ as a contraction and $V$ as an isometry satisfying $T^*V=VT^*$, then there exists a unique pair of doubly commuting isometries on the minimal isometric dilation space of $T$. As an application, we provide a new proof for pairs of commuting operators consisting of an isometry and a co-isometry are doubly commuting.