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.