Three results in linear dynamics

TL;DR

This paper presents three key results in linear dynamics: limitations on supercyclic operators in certain spaces, computation of a crucial constant for weighted shifts, and an answer to an open question on supercyclic vectors.

Contribution

It introduces new constraints on supercyclic operators, calculates a specific constant for weighted shifts, and resolves an open problem regarding supercyclic vectors.

Findings

No strongly supercyclic weighted composition operators on $H(D)$

Computed the constant $$ for weighted backward shifts on $ell^p$ and $c_0$

Confirmed the existence of supercyclic vectors in a specific context

Abstract

In this article, first we show that the Fr\'echet space $H(\Bbb D)$ cannot support strongly supercyclic weighted composition operators. Then we compute the constant $\epsilon$ for weighted backward shifts on $\ell^p$ ($1\le p<\infty$) and $c_0$. This constant is used to find strongly hypercyclic scalar multiples of non-invertible strongly supercyclic Banach space operators. Finally, we give an affirmative answer to a recent open question concerning supercyclic vectors.