Linear resolvent growth of a weak contraction does not imply its similarity to a normal operator

It was shown in [1] that if T is a contraction in a Hilbert space with finite defect (i.e., //T// less than or equal to 1 and rank(I - T*T) < infinity), and if the spectrum sigma (T) does not coincide with the closed unit disk D, then the Linear Resolvent Growth condition //(lambdaI-T_(-1)// less than or equal to (C)/(dist(lambda,sigma (T))), lambda is an element of C\sigma (T) implies that T is similar to a normal operator. The condition rank(I - T*T) < infinity measures how close T is to a unitary operator. A natural question is whether this condition can be relaxed. For example, it was conjectured in [1] that this condition can be replaced by the condition I - T*T is an element of G(1) where G(1) denotes the trace class. In this note we show that this conjecture is not true, and that, in fact, one cannot replace the condition rank(I - T*T) < infinity by any reasonable condition of closeness to a unitary operator.