Operators with Inverses Similar to Their Adjoints
U. N. Singh, Kanta Mangla · Proceedings of the American Mathematical Society · 1973
If $T$ is an invertible operator on a Hilbert space such that ${S^{ - 1}}{T^{ - 1}}S = {T^ \ast }$ and $0 otin {\text {Cl}}(W(S))$ for some invertible operator $S$, where ${\text {Cl}}(W(S))$ denotes the closure of the numerical range of $S$ and ${T^ \ast }$ is the adjoint of $T$, then it is shown that $T$ is similar to a unitary operator. In fact, this has been proved as a corollary to a more general result, which also includes the corresponding result of J. P. Williams for selfadjoint operators.