ACTA issues

Operators similar to partial isometries

C. Badea, M. Mbekhta

Acta Sci. Math. (Szeged) 71:3-4(2005), 663-680

Abstract. A Hilbert space bounded linear operator is said to be regular if its range is closed and its kernel is included in the intersection of the ranges of all iterates. We prove that if $T$ is regular, then $T$ is similar to a partial isometry if and only if $T$ is power bounded and there exists a power bounded operator $S$ such that $TST =T$. This is a generalization of a similarity criterion due to B. Sz.-Nagy. The regularity condition cannot be avoided. Indeed, using Pisier's example of a polynomially bounded operator not similar to a contraction, two polynomially bounded operators $T$ and $S$ are constructed such that $TST = T$ and $STS =S$, but $T$ is not similar to a partial isometry.

AMS Subject Classification (1991): 47A05, 47A10, 47B47

Keyword(s): partial isometries, similarity problems, generalized inverse, generalized spectrum, the commutator equation

Received April 28, 2005, and in final form September 14, 2005. (Registered under 5893/2009.)