Operators Similar to Contractions and Their Similarity to a Normal Operator

It has been established quite recently [1], that a contraction T , having finite defects (rank (I − T T ), rank (I − TT ) < ∞) and the spectrum σ(T ) not filling in the closed unit disk D, is similar to a normal operator if and only if C1(T ) = sup λ6∈σ(T ) ||(T − λ)|| · dist(λ, σ(T )) < ∞. It was shown later [9], that the “only if” part of the assertion is no longer true if T is not a contraction. It is natural to ask what larger (than the class of contractions) classes of operators obey the linear resolvent growth test, formulated above. In this paper, we obtain results of the same type for operators that are similar to a contraction. Their proofs are based on a refined version of a theorem obtained in [6].