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

## S. Kupin and S. Treil

It was shown by N. Nikolski and N. Benamara that if T is a contraction in a Hilbert space with finite defect (), and its spectrum  doesn't coincide with the closed unit disk , then the following Linear Resolvent Growth condition

implies that T is similar to a normal operator.

The condition  characterizes how close is T to a unitary operator. A natural question arises about relaxing this condition. For example, it was conjectured that one can replace the condition  by , where  denotes the trace class.

In this note we show that this conjecture is not true, moreover it is impossible to replace the condition  by any reasonable condition of closedness to a unitary operator. For example, we construct a contraction T (i. e. ), , satisfying , where  stands for the Schatten-von-Neumann class, satisfying the above Linear Resolvent Growth condition but not similar to a normal operator.

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Serguei Treil

9/17/1999