## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 45, Number 1 (2001), 229-242.

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

#### Abstract

It was shown in \cite{NB} that if $T$ is a contraction in a Hilbert space with finite defect (i.e., $\|T\|\le 1$ and $\operatorname{rank} (I- T^*T) <\infty$), and if the spectrum $\sigma(T)$ does not coincide with the closed unit disk $\overline{\mathbb{D}}$, then the Linear Resolvent Growth condition $$ \|(\la I - T)^{-1} \|\le\frac{C}{\operatorname{dist}(\la,\si(T))},\ \la\in\bc\backslash \si(T) $$ implies that $T$ is similar to a normal operator. The condition $\operatorname{rank}(I - T^*T)<\infty$ measures how close $T$ is to a unitary operator. A natural question is whether this condition can be relaxed. For example, it was conjectured in \cite{NB} that this condition can be replaced by the condition $I - T^*T\in \fS_1$, where $\fS_1$ denotes the trace class. In this note we show that this conjecture is not true, and that, in fact, one cannot replace the condition $\operatorname{rank}(I - T^*T)<\infty$ by any reasonable condition of closeness to a unitary operator.

#### Article information

**Source**

Illinois J. Math., Volume 45, Number 1 (2001), 229-242.

**Dates**

First available in Project Euclid: 13 November 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1258138265

**Digital Object Identifier**

doi:10.1215/ijm/1258138265

**Mathematical Reviews number (MathSciNet)**

MR1849996

**Zentralblatt MATH identifier**

0994.47006

**Subjects**

Primary: 47A10: Spectrum, resolvent

Secondary: 47A11: Local spectral properties 47A45: Canonical models for contractions and nonselfadjoint operators 47B15: Hermitian and normal operators (spectral measures, functional calculus, etc.)

#### Citation

Kupin, S.; Treil, S. Linear resolvent growth of a weak contraction does not imply its similarity to a normal operator. Illinois J. Math. 45 (2001), no. 1, 229--242. doi:10.1215/ijm/1258138265. https://projecteuclid.org/euclid.ijm/1258138265