Relations between principal functions of p-hyponormal operators

Abstract

Let $T=U\left|T\right|$ be a bounded linear operator with the associated polar decomposition on a separable infinite dimensional Hilbert space. For , let $T_t=\left|T{|}^{t}U\right|T{|}^{1-t}$ and $g_T$ and $g_{T_t}$ be the principal functions of $T$ and $T_t$, respectively. We show that, if $T$ is an invertible semi-hyponormal operator with trace class commutator $\left[\right|T|,U]$, then $g_T=g_{T_t}$ almost everywhere on $\mathbf{C}$. As a biproduct we reprove Berger's theorem and index properties of invertible $p$-hyponormal operators.