(untitled) (Communications in Mathematical Research, 2024)
An operator T on a separable, infinite dimensional, complex Hilbert space H is called conjugate normal if C|T|C = |T^∗| for some conjugate linear, isometric involution C on H. This paper focuses on the invariance of conjugate normality under similarity. Given an operator T, we prove that every operator A similar to T is conjugate normal if and only if there exist complex numbers λ_1, λ_2 such that (T−λ_1)(T−λ_2)=0.