Chaos and dF-semi-transitivity of operators on Banach C*-modules

TL;DR

This paper characterizes various forms of chaos and transitivity for operators on Banach C*-modules, introducing new notions and providing concrete examples and characterizations in non-commutative L2-spaces.

Contribution

It introduces new chaos concepts like D0-Devaney chaos and disjoint dF-D-transitivity, offering comprehensive characterizations and examples in the context of operators on Banach C*-modules.

Findings

Characterized Li-Yorke chaos for weighted shift operators via operator-valued weights.

Established equivalence of Li-Yorke chaos for shifts with fixed operator W on Hilbert space.

Constructed examples of chaotic operators with specific transitivity properties.

Abstract

In this paper, we characterize Li-Yorke chaotic generalized weighted shift operators on the standard Hilbert module over the C*-algebra of compact operators on a separable Hilbert space in terms of operator-valued weights of these shifts. Also, we prove that if all the weights are equal to a fixed operator W on a Hilbert space H, then the induced generalized weighted shift on the Hilbert C*-module is Li-Yorke chaotic if and only if W is Li-Yorke chaotic on H. Moreover, we construct an example of a Li-Yorke chaotic non topologically transitive generalized weighted shift on the Hilbert C*-module. Next, we introduce a new notion of Devaney chaos, which we call D0-Devaney chaos, and we completely characterize in terms of the weight functions Devaney chaotic and D0-Devaney chaotic adjoints of weighted composition operators acting on the space of Radon measures. As an application, we construct D0 Devaney chaotic operators that are not Devaney chaotic in the classical sense and vice versa. Also, we introduce a new concept of disjoint dF-D-transitivity. We completely characterize in terms of the weight functions dF-D-transitive and dF-(semi-)transitive adjoints of weighted composition operators acting on weighted spaces of Radon measures. We construct examples of dF-D-transitive operators that are not dF-semi-transitive and vice versa. Also, in the examples, we illustrate the differences between dF-(semi-)transitivity on weighted and non-weighted spaces of Radon measures. At the end, we characterize Devaney chaotic elementary operators on non-commutative L2-spaces, and we provide concrete examples. Also, we construct a topologically mixing, non-Devaney chaotic operator on noncommutative L2-space.