On p-quasi-hyponormal operators

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∗∣2p)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 H0(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.