Sets of essentially unitary operators

Abstract:

Let ${U_e}$ be the set of essentially unitary operators on a separable Hilbert space $H$; for $1 \leqslant p \leqslant \infty$, let ${U_p}$ be the set of operators $T$ such that $I - T^{\ast }T$ lies in the Schatten $p$-ideal and the spectrum of $T$ does not fill the unit disc; and let $U_e^n$ be the set of operators in ${U_e}$ of Fredholm index $n$. The author proves that each $U_e^n$ is closed and path connected, that ${U_p}$ is dense in ${U_e}^0$ and ${U_p}$ is path connected for each $p$, and that all these sets are invariant under Cayley transform. It is proved that the spectrum is continuous on ${U_\infty }$ but not on ${U_e}$, while the spectral radius is continuous on ${U_e}$. Sufficient conditions that an operator in ${U_e}$ have a nontrivial hyperinvariant subspace are given, and it is proved that the general hyperinvariant subspace problem can be reduced to that problem for perturbations of the bilateral shift. The product of commuting operators in ${U_p}$ is ${U_p}$, but this result is false in general. Quasisimilarity in ${U_e}$ is also studied; quasisimilar operators in ${U_e}\backslash {U_\infty }$ are unitarily equivalent modulo the ideal of compacts, and this result also holds in ${U_\infty }$ if the spectrum is also preserved.