Subspace-diskcyclic sequences of linear operators

TL;DR

This paper investigates conditions under which sequences of bounded linear operators exhibit diskcyclicity and subspace-diskcyclicity, extending the concept to dense orbits within subspaces of Banach and Hilbert spaces.

Contribution

It introduces new criteria for diskcyclicity and subspace-diskcyclicity of operator sequences, generalizing previous results and providing a framework for understanding their dense orbit properties.

Findings

Established conditions for diskcyclicity of operator sequences.

Developed subspace-diskcyclicity criteria analogous to existing theorems.

Provided examples illustrating the application of these criteria.

Abstract

A sequence $\{T_n\}_{n=1}^{\infty}$ of bounded linear operators between separable Banach spaces $X, Y$ is called diskcyclic if there exists a vector $x\in X$ such that the disk-scaled orbit $\{\alpha T_n x: n\in \mathbb{N}, \alpha \in\mathbb{C}, | \alpha | \leq 1\}$ is dense in $Y$. In the first section of this paper we study some conditions that imply the diskcyclicity of $\{T_n\}_{n=1}^{\infty}$. In particular, a sequence $\{T_n\}_{n=1}^{\infty}$ of bounded linear operators on separable infinite dimensional Hilbert space $\mathcal{H}$ is called subspace-diskcyclic with respect to the closed subspace $M\subseteq \mathcal{H},$ if there exists a vector $x\in \mathcal{H}$ such that the disk-scaled orbit $\{\alpha T_n x: n\in \mathbb{N}, \alpha \in\mathbb{C}, | \alpha | \leq 1\}\cap M$ is dense in $M$. In the second section we survey some conditions and subspace-diskcyclicity criterion (analogue the results obtained by the some mathematicians in \cite{MR2261697, MR2720700, MR1111569}) which are sufficient for the sequence $\{T_n\}_{n=1}^{\infty}$ to be subspace-diskcyclic.