The spectra of unbounded hyponormal operators

Abstract

A bounded operator T T on a Hilbert space is said to be completely hyponormal if T ∗ T − T T ∗ ≧ 0 {T^\ast }T - T{T^\ast } \geqq 0 and if T T has no nontrivial reducing space on which it is normal. If 0 0 is in the spectrum of such an operator T T and if the spectrum of T T near 0 0 is not “too dense,” then the unbounded operator T − 1 {T^{ - 1}} acts as though it were bounded. In particular, under certain conditions, T − 1 {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 {T^{ - 1}} onto the coordinate axes.