Conditions Implying Self-adjointness and Normality of Operators

Abstract

In this paper, we give new characterizations of self-adjoint and normal operators on a Hilbert space \(\mathcal {H}\). Among other results, we show that if \(\mathcal {H}\) is a finite-dimensional Hilbert space and \(T\in \mathfrak {B}(\mathcal {H})\), then T is self-adjoint if and only if there exists \(p>0\) such that \(|T|^p\le |\textrm{Re}\,(T)|^p\). If in addition, T and \(\textrm{Re}\,T\) are invertible, then T is self-adjoint if and only if \(\log \,|T|\le \log \,|\textrm{Re}\,(T)|\). Considering the polar decomposition \(T=U|T|\) of \(T\in \mathfrak {B}(\mathcal {H})\), we show that T is self-adjoint if and only if T is p-hyponormal (log-hyponormal) and U is self-adjoint. Also, if \(T=U|T|\in \mathfrak {B}({\mathcal {H}})\) 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.