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

If $A$ is a bounded linear operator on a Hilbert space, define ${r_e}(A)$, the essential spectral radius of $A$, by

$${r_e}(A): = \sup \{ \vert\lambda \vert:\lambda \in \operatorname{ess}(A) = \operatorname{essential}\;\operatorname{spectrum}\;{\text{of}}\;A\} .$$

It is shown that

$${r_e}(A) = \operatorname{inf}\{ \alpha ({S^{ - 1}}AS)\vert S:H \t... ...;{\text{bounded}}\;{\text{invertible}}\;{\text{linear}}\;\operatorname{map}\} ,$$

where $\alpha$ is the Kuratowski measure of noncompactness. As a consequence, a charcterization of the similarity of a linear strict set-contraction is obtained.