Operators with Powers Essentially Similar to Those of Their Adjoints
S. M. Patel · Proceedings of the American Mathematical Society · 1974
Let T be an operator on a Hilbert space H. In the present note the following result is obtained: If T is an operator such that for some integers $p,q,S{T^{ \ast p}} = {T^q}S + K$, where 0 is not in the essential numerical range of S, and K is compact, then for any complex number $\lambda$ in the essential spectrum of T, ${\lambda ^{ \ast p}} = {\lambda ^q}$.