Linear resolvent growth test for similarity of a weak contraction to a normal operator

Abstract

It is proved in Benamara-Nikolski [1] that if the spectrum σ(T) of a contraction T with finite defects (rank(I−T^*T)=rank (I−TT^*)<∞) does not coincide with $\bar D$ , then the contraction is similar to a normal operator if and only if $C_1 (T) = \mathop {\sup }\limits_{\lambda \in C\backslash \sigma (T)} \parallel (T - \lambda )^{ - 1} \parallel dist(\lambda ,\sigma (T))< \infty .$

The examples of Kupin-Treil [9] show that the result is no longer true if we replace the condition rank (I−T^*T)<∞ by its weakened versionG, whereG denotes the class of nuclear operators.

We prove in this paper that, however, the following theorem holds