Spectral properties of p-hyponormal operators

Abstract

Let ℋ be a complex Hilbert space and B(ℋ) be the algebra of all bounded linear opeators on ℋ. An operator T ∈ B(ℋ) is said to be p-hyponormal if (T*T)p–(TT*)p. If p = 1, T is hyponormal and if p = ½ is semi-hyponormal. It is well known that a p-hyponormal operator is p-hyponormal for q≤p. Hyponormal operators have been studied by many authors. The semi-hyponormal operator was first introduced by D. Xia in [7]. The p-hyponormal operators have been studied by A. Aluthge in [1]. Let T be a p-hyponormal operator and T=U|T| be a polar decomposition of T. If U is unitary, Aluthge in [1] proved the following properties.