Frequent, disjoint hypercyclicity and strong topological transitivity of generalized weighted shift operators on Hilbert C-modules

TL;DR

This work studies dynamical properties of the generalized bilateral weighted shift $T_{U,W}$ on the Hilbert $C$-module $\ell_{2}(\mathcal{A})$, focusing on the Frequent Hypercyclicity Criterion, chaos, disjoint hypercyclicity, and $\mathcal{F}$-transitivity via Furstenberg families. It provides a robust FH Criterion-based framework yielding sufficient conditions for frequence hypercyclicity and chaos, and shows that for $U=I$ the operator $T_{W}$ is chaotic if and only if it satisfies the FH Criterion, with chaos implying mixing. In the multivariable setting, the authors characterize densely disjoint hypercyclicity for tuples $\big(T_{U^{(s)},W^{(s)}}\big)$ under a cross-orthogonality condition on the unitary parts and give explicit constructions where all weights are invertible. The paper further develops $\mathcal{F}$-transitivity criteria for $T_{U,W}$, establishing uniform $\mathcal{F}$-convergence on dense sets and providing methods to obtain $\mathcal{F}$-transitive and mixing behavior, including a topologically mixing example via a fixed weight operator. Overall, the results extend classical weighted-shift dynamics from $\ell^{p}$-spaces to Hilbert $C$-modules and illuminate how frequent recurrence properties can be realized in this broader operator-class context.

Abstract

In this paper we study some dynamical properties such as Frequent Hypercyclicity Criterion, chaos, disjoint hypercyclicity and F-transitivity via Furstenberg family F for generalized bilateral weighted shift operator on the standard Hilbert C-module over C-algebra of compact operators on a separable Hilbert space.

Theorems & Definitions (19)

• Theorem 2.1
• Proposition 2.2
• Definition 2.3
• Proposition 3.1
• proof
• Example 3.2
• Theorem 3.3
• proof
• Theorem 3.4
• proof