Regular covariant representations and their Wold-type decomposition

TL;DR

This paper extends the Shimorin-Wold-type decomposition to regular, completely bounded covariant representations with certain growth conditions, introducing new concepts and analyzing shifts in a non-commutative setting.

Contribution

It generalizes the Wold-type decomposition for a broader class of covariant representations, introducing the concepts of regularity, algebraic core, and reduced minimum modulus.

Findings

Extended Wold-type decomposition to regular, completely bounded covariant representations.

Introduced the concepts of regular, algebraic core, and reduced minimum modulus.

Analyzed weighted unilateral and bilateral shifts in a non-commutative framework.

Abstract

Olofsson introduced a growth condition regarding elements of an orbit for an expansive operator and generalized Richter's wandering subspace theorem. Later on, using the Moore-Penrose inverse, Ezzahraoui, Mbekhta, and Zerouali extended the growth condition and obtained a Shimorin-Wold-type decomposition. Shimorin-Wold-type decomposition for completely bounded covariant representations, which are close to isometric representations, is obtained in \cite{HV19}. This paper extends this decomposition for regular, completely bounded covariant representation having reduced minimum modulus $\geq 1$ that satisfies the growth condition. To prove the decomposition, we introduce the terms regular, algebraic core, and reduced minimum modulus in the completely bounded covariant representation setting and work out several fundamental results. Consequently, we shall analyze the weighted unilateral shift introduced by Muhly and Solel and introduce and explore a non-commutative weighted bilateral shift.