Linear resolvent growth of a weak contraction does not imply its similarity to a normal operator (Q5954265)

Let \(T\) be a Hilbert space contraction satisfying the linear resolvent growth condition NEWLINE\[NEWLINE\|(\lambda I- T)^{-1}\|\leq C/\text{dist}(\lambda, \sigma(T)),\quad \lambda\in \mathbb{C}\setminus\sigma(T)NEWLINE\]NEWLINE and whose spectrum \(\sigma(T)\) does not coincide with the closed unit disk.NEWLINENEWLINENEWLINE\textit{N. E. Benamara} and \textit{N. K. Nikolski} [Proc. London Math. Soc., 78, 585-626 (1999; Zbl 1028.47500)] showed that if \(T\) has finite defect, i.e. \(\text{rank}(I- T^* T)<\infty\), then \(T\) is similar to a normal operator. They also conjectured that the condition ``finite defect'' can be weakened to \(I- T^* T\in C_1\) (the trace class). The present paper provides a counterexample to this hypothesis. At the same time, the authors end the paper with a modified version of the above conjecture. Namely, they conjecture that the condition NEWLINE\[NEWLINE\sup\{\|I- T^*_\mu T_\mu\|_1:|\mu|\leq 1\}<\inftyNEWLINE\]NEWLINE implies similarity to a normal operator. Here \(T_\mu= (T- \mu I)(I-\overline\mu T)^{-1}\).