On p-quasi-hyponormal operators

Abstract

A Hilbert space operator A ∈ B( H) is said to be p-quasi-hyponormal for some 0 < p ⩽ 1, A ∈ p − QH, if A ∗(∣ A∣ 2p − ∣ A ∗∣ 2 p ) A ⩾ 0. If H is infinite dimensional, then operators A ∈ p − QH are not supercyclic. Restricting ourselves to those A ∈ p − QH for which A −1(0) ⊆ A ∗-1(0), A ∈ p ∗ − QH, a necessary and sufficient condition for the adjoint of a pure p ∗ − QH operator to be supercyclic is proved. Operators in p ∗ − QH satisfy Bishop’s property ( β). Each A ∈ p ∗ − QH has the finite ascent property and the quasi-nilpotent part H 0( A − λI) of A equals ( A − λI) -1(0) for all complex numbers λ; hence f( A) satisfies Weyl’s theorem, and f( A ∗) satisfies a-Weyl’s theorem, for all non-constant functions f which are analytic on a neighborhood of σ( A). It is proved that a Putnam–Fuglede type commutativity theorem holds for operators in p ∗ − QH.