The spectra of unbounded hyponormal operators

Abstract:

A bounded operator $T$ on a Hilbert space is said to be completely hyponormal if ${T^\ast }T - T{T^\ast } \geqq 0$ and if $T$ has no nontrivial reducing space on which it is normal. If $0$ is in the spectrum of such an operator $T$ and if the spectrum of $T$ near $0$ is not “too dense,” then the unbounded operator ${T^{ - 1}}$ acts as though it were bounded. In particular, under certain conditions, ${T^{ - 1}}$ has a rectangular representation with absolutely continuous real and imaginary parts whose spectra are the closures of the projections of the spectrum of ${T^{ - 1}}$ onto the coordinate axes.