# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Perturbations of operators similar to contractions and the commutator equation

### Tom 150 / 2002

Studia Mathematica 150 (2002), 273-293 MSC: Primary 47A50; Secondary 47B47, 47A55. DOI: 10.4064/sm150-3-5

#### Streszczenie

Let $T$ and $V$ be two Hilbert space contractions and let $X$ be a linear bounded operator. It was proved by C. Foiaş and J. P. Williams that in certain cases the operator block matrix $R(X;T,V)$ (equation (1.1) below) is similar to a contraction if and only if the commutator equation $X = TZ-ZV$ has a bounded solution $Z$. We characterize here the similarity to contractions of some operator matrices $R(X;T,V)$ in terms of growth conditions or of perturbations of $R(0;T,V)=T\oplus V$.