On p-quasi-hyponormal operators Original Research Article

A Hilbert space operator A set membership, variant B(H) is said to be p-quasi-hyponormal for some 0 < p less-than-or-equals, slant 1, A set membership, variant p − QH, if A*(midAmid2p − midA*mid2p)A greater-or-equal, slanted 0. If H is infinite dimensional, then operators A set membership, variant p − QH are not supercyclic. Restricting ourselves to those A set membership, variant p − QH for which A−1(0) subset of or equal to A*-1(0), A set membership, variant 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 set membership, variant 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.