Purpose Since weighted shifts play a vital role in linear dynamics so in 2021 Chan & Sanders considered weighted shift operators on ℓp (Z) that are not hypercyclic and proved that under certain conditions they can be factorized as products of hypercyclic shifts. The purpose of this article is to extend their results to operator weighted shifts on ℓ2 (K) by using the concept of generalized shift of higher multiplicity developed herein. Design/methodology/approach Traditionally, an operator T is considered as a shift on a space H if there is some canonical basis for H such that T shifts every basis vector to the immediate next basis vector, maybe with some weight attached to it. This clearly indicates that an operator which is a shift with respect to a basis for H may not remain a shift if the basis is changed. This motivates the definition of a generalized shift operator. We begin with the case of multiplicity one and then extend it to higher finite multiplicity. We then develop the idea so that we can frame conditions under which a bilateral shift can be factorized as product of hypercyclic generalized shifts. Findings A generalized bilateral backward weighted shift (GBBWS) is defined in terms of a bijection on Z and it is shown that if there are two bijections σ and ρ which generate the same generalized shift, then there exists a unique c such that σ (i) = ρ (i + c) for all integers i. We determine conditions under which the direct sum of generalized shifts is again a generalized shift. We also show that for a uniformly bounded sequence of invertible diagonal operators Ai on a separable complex Hilbert space K of finite dimension, if W is bilateral backward weighted shift on ℓ2 (K) with weight sequence Ai then there exists hypercyclic GBBWS T and P on ℓ2 (K) such that W = TP. Originality/value The idea of generalized shift of multiplicity one was introduced by Chan & Sanders in 2018. However, we have developed the idea further, particularly extending it to the case of shifts of higher multiplicity. The factorization of shifts as product of hypercyclic shifts was also done by them. We have extended their result to higher dimension, and our approach to the problem is completely different from theirs.
Hazarika et al. (Tue,) studied this question.