Weak Sensitive Compactness for Linear Operators

Abstract

Let $(X,T)$ be a linear dynamical system, where $X$ is a separable Banach space and $T:X\to X$ is a bounded linear operator. We show that if $(X,T)$ is invertible, then $(X,T)$ is weakly sensitive compact if and only if $(X,T)$ is thickly weakly sensitive compact; and that there exists a system $(X\times Y,T\times S)$ such that:

(1)	$(X\times Y,T\times S)$ is cofinitely weakly sensitive compact;
(2)	$(X,T)$ and $(Y,S)$ are weakly sensitive compact; and
(3)	$(X,T)$ and $(Y,S)$ are not syndetically weakly sensitive compact.

We also show that if $(X,T)$ is weakly sensitive compact, where $X$ is a complex Banach space, then the spectrum of $T$ meets the unit circle.