Essential Normality of Operators Close to Isometries

Abstract

Let p(z, w) denote a complex polynomial in non-commutative variables z and w. Let ${\fancyscript{F}}$ be a collection of Hilbert-space operators such that the following hold: ${\fancyscript{F}}$ is invariant under unital *-representations; if ${N \in \fancyscript{F}}$ is invertible then p(N, N*) = 0. Then p(T, T*) is compact for every Fredholm member T of ${\fancyscript{F}}$. We use the above scheme to deduce essential normality of several operators close to isometries. A sample result reads as follows: A finitely multi-cyclic, expansive m-isometry is a compact perturbation of an essentially normal isometry. This result is best possible in the sense that there exists a cyclic 3-isometry which is not even essentially normal.