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 fini te 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 dis k D, then the Linear Resolvent Growth condition //(lambdaI-T_(-1)// less than or equal to (C)/(dist(lambda,sigma (T))), lam bda 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 cl ass. In this note we show that this conjecture is not true, and that, in fa ct, one cannot replace the condition rank(I - T*T) < infinity by any reason able condition of closeness to a unitary operator.