Structure on Powers of p-Hyponormal and log-Hyponormal Operators

Abstract.

Inspired by the problem of powers of hyponormal operators, this paper is to discuss the structure on powers of p-hyponormal and log-hyponormal operators. The structure on powers of operators consists of same-side structure and different-side structure. The same-side structure means relations between $T^{{*}^{n+m}} T^{n+m}$ and $T^{{*}^{n}} T^n ({\rm or} T^nT^{{*}^{n}}\,{\rm and}\,T^{n+m}T^{{*}^{n+m}})$ , and the different-side structure means relations between $T^{{*}^{m}} T^{m}\,{\rm and}\,T^{n}T^{{*}^{n}}$ where m, n are positive integers and T is a bounded linear operator on a Hilbert space. Thus, the original problem of powers of hyponormal operators belongs to different-side structure on powers of hyponormal operators. The structure on powers of p-hyponormal operators for p > 0 is emphasized. Also, some applications are obtained.