Fast orbital convergence reveals more hypercyclic vectors
Submitted: 2022-11-28
|Accepted: 2025-03-28
|Published: 2025-10-06
Copyright (c) 2025 T. K. S. Moothathu
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
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Keywords:
Banach space, hypercyclic operator, orbital speed, frequent hypercyclicity
Supporting agencies:
Abstract:
Let X be an infinite dimensional separable Banach space, T : X → X be a hypercyclic operator, and x ∈ X be a (frequently) hypercyclic vector of T. We show that if the terms from the T-orbit of x converge to a vector y sufficiently fast, then y is also a hypercyclic vector of T. As a corollary, we deduce that if T is a frequently hypercyclic operator with spectral radius r(T) = 1, then lim_{n\to \infty} ∥Tnx∥^{1/n} = 1 for every frequently hypercyclic vector x of T. Some related observations are also made.
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