Weak Sensitive Compactness for Linear Operators

Let [Formula: see text] be a linear dynamical system, where [Formula: see text] is a separable Banach space and [Formula: see text] is a bounded linear operator. We show that if [Formula: see text] is invertible, then [Formula: see text] is weakly sensitive compact if and only if [Formula: see text] is thickly weakly sensitive compact; and that there exists a system [Formula: see text] such that:

(1) [Formula: see text] is cofinitely weakly sensitive compact; (2) [Formula: see text] and [Formula: see text] are weakly sensitive compact; and (3) [Formula: see text] and [Formula: see text] are not syndetically weakly sensitive compact.

We also show that if [Formula: see text] is weakly sensitive compact, where [Formula: see text] is a complex Banach space, then the spectrum of [Formula: see text] meets the unit circle.