Conditions Implying Self-adjointness and Normality of Operators

In this paper, we give new characterizations of self-adjoint and normal operators on a Hilbert space H. Among other results, we show that if H is a finite-dimensional Hilbert space and T is an element of B(H), then T is self-adjoint if and only if there exists p > 0 such that |T|(p )<= | Re(T)|(p). If in addition, T and ReT are invertible, then T is self-adjoint if and only if log|T| <= log|Re(T)|. Considering the polar decomposition T = U|T| of T is an element of B(H), we show that T is self-adjoint if and only if T is p-hyponormal (log-hyponormal) and U is self-adjoint. Also, if T = U|T| is an element of B(H) is a log-hyponormal operator and the spectrum of U is contained within the set of vertices of a regular polygon, then T is necessarily normal.