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Hypercyclic Operators on Hilbert C*-modules

Stefan Ivkovic

TL;DR

The paper analyzes hypercyclicity and chaos for generalized bilateral weighted shift operators on the standard Hilbert $C^{*}$-module over the compact operators $B_0(H)$ on a separable Hilbert space. It introduces the operator $T_{U,W}$ with explicit norm bounds and inverse, derives formulas for iterates, and proves a dense hypercyclicity criterion equivalent to a constructive sequence condition involving finite-rank projections. Extending these ideas, it develops hypercyclicity criteria for operators on non-unital $C^{*}$-algebras, showing equivalence between hypercyclicity and the existence of approximating sequences satisfying forward/backward contraction properties under automorphisms $\Phi$. The results are illustrated with concrete examples and connected to classical settings through unitary conjugation and $C_0(X)$-type algebras, thereby broadening linear dynamics to Hilbert $C^{*}$-modules and non-unital algebras. Overall, the work provides a framework to characterize, construct, and understand chaotic dynamics in operator-algebra contexts.

Abstract

In this paper, we characterize hypercyclic generalized bilateral weighted shift operators on the standard Hilbert module over the C*-algebra of compact operators on the separable Hilbert space. Moreover, we give necessary and sufficient conditions for these operators to be chaotic and we provide concrete examples. In addition, we characterize a class of hypercyclic operators on non-unital C*-algebras and we provide concrete examples.

Hypercyclic Operators on Hilbert C*-modules

TL;DR

The paper analyzes hypercyclicity and chaos for generalized bilateral weighted shift operators on the standard Hilbert -module over the compact operators on a separable Hilbert space. It introduces the operator with explicit norm bounds and inverse, derives formulas for iterates, and proves a dense hypercyclicity criterion equivalent to a constructive sequence condition involving finite-rank projections. Extending these ideas, it develops hypercyclicity criteria for operators on non-unital -algebras, showing equivalence between hypercyclicity and the existence of approximating sequences satisfying forward/backward contraction properties under automorphisms . The results are illustrated with concrete examples and connected to classical settings through unitary conjugation and -type algebras, thereby broadening linear dynamics to Hilbert -modules and non-unital algebras. Overall, the work provides a framework to characterize, construct, and understand chaotic dynamics in operator-algebra contexts.

Abstract

In this paper, we characterize hypercyclic generalized bilateral weighted shift operators on the standard Hilbert module over the C*-algebra of compact operators on the separable Hilbert space. Moreover, we give necessary and sufficient conditions for these operators to be chaotic and we provide concrete examples. In addition, we characterize a class of hypercyclic operators on non-unital C*-algebras and we provide concrete examples.
Paper Structure (4 sections, 8 theorems, 103 equations)

This paper contains 4 sections, 8 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalized weighted bilateral shift operators over $C^{*}$-algebras
  4. Hypercyclic operators on $C^{*}$-algebras

Key Result

Proposition 3.1

Let $(t_n)_n$ be an unbounded sequence of nonnegative integers. We denote $T_{U,W,n}:=T_{U,W}^{t_n}$ for all $n\in\mathbb{N}$. Then, the followings are equivalent:

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Example 3.3
  • Proposition 3.4
  • proof
  • ...and 14 more