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On Certain forms of Transitivities for Linear Operators

Nayan Adhikary, Anima Nagar

Abstract

In this article we give several characterizations for various transitivity properties for linear operators. We define a general form of `Hypercyclicity Criterion' using a Furstenberg family $\mathcal{F}$ to characterize $\mathcal{F}$-transitive operators. In particular, we find an equivalent characterization for mixing operators. We study proximal and asymptotic relations for linear operators and prove that the difference between mixing operators and Kitai's Criterion can be presented through these relations. Finally, we find an equivalent characterization of strongly transitive abd strongly product transitive operators.

On Certain forms of Transitivities for Linear Operators

Abstract

In this article we give several characterizations for various transitivity properties for linear operators. We define a general form of `Hypercyclicity Criterion' using a Furstenberg family to characterize -transitive operators. In particular, we find an equivalent characterization for mixing operators. We study proximal and asymptotic relations for linear operators and prove that the difference between mixing operators and Kitai's Criterion can be presented through these relations. Finally, we find an equivalent characterization of strongly transitive abd strongly product transitive operators.

Paper Structure

This paper contains 4 sections, 9 theorems, 13 equations.

Table of Contents

  1. Introduction:
  2. Preliminaries
  3. On $\mathcal{F}$-Transitive Operators
  4. Strongly Transitive Operators

Key Result

Theorem 2.6

A linear dynamical system is either sensitive or equicontinuous.

Theorems & Definitions (21)

  • Theorem 2.6
  • proof
  • proof
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.10
  • proof
  • Theorem 3.11
  • proof
  • ...and 11 more