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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
we construct a contraction T (i. e. ), ,
stands for the Schatten-von-Neumann class, satisfying the above Linear
Resolvent Growth condition but not similar to a normal operator.