Subspace-diskcyclic sequences of linear operators

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 $\{\alpha T_{n}x:n\in \mathbb{N},\alpha \in \mathbb{C},|\alpha |\le 1\}$ 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 $\{\alpha T_{n}x:n\in \mathbb{N},\alpha \in \mathbb{C},|\alpha |\le 1\}\cap M$ 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.

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• J-class sequences of linear operators