Similarity of a linear strict set-contraction and the radius of the essential spectrum

Abstract:

If $A$ is a bounded linear operator on a Hilbert space, define ${r_e}(A)$, the essential spectral radius of $A$, by \[ {r_e}(A): = \sup \{ |\lambda |:\lambda \in \operatorname {ess}(A) = \operatorname {essential}\;\operatorname {spectrum}\;{\text {of}}\;A\} .\] It is shown that \[ {r_e}(A) = \operatorname {inf}\{ \alpha ({S^{ - 1}}AS)|S:H \to H\;{\text {is}}\;{\text {a}}\;{\text {bounded}}\;{\text {invertible}}\;{\text {linear}}\;\operatorname {map}\} ,\] where $\alpha$ is the Kuratowski measure of noncompactness. As a consequence, a charcterization of the similarity of a linear strict set-contraction is obtained.