p -Hyponormal Operators and Quasisimilarity

Abstract

In this paper it is shown that the normal parts of quasisimilar p-hyponormal operators are unitarily equivalent, a p-hyponormal operator compactly quasisimilar to an isometry is normal, and a p-hyponormal spectral operator is normal.

A generalized commutativity theorem for pk-quasihyponormal operators

“…Since X is a quasiaffinity, and since 0 / ∈ σ p (A) =⇒ 0 / ∈ σ(A), the hypothesis AB (X) = 0 implies that A is an invertible phyponormal operator such that δ A −1 B (X) = 0. Applying [7,Theorem 11], it follows that A −1 is normal, B is a scalar operator similar to A −1 and δ A * −1 B * (X) = 0 = A * B * (X).…”

“…The (classical) Putnam-Fuglede commutativity theorem says that if A and B are normal operators, then δ −1 AB (0) ⊆ δ −1 A * B * (0). Over the years, the Putnam-Fuglede commutativity theorem has been extended to various classes of operators, each more general than the class of normal operators, and to the elementary operator AB to prove that −1 AB (0) ⊆ −1 A * B * (0) for many of these classes of operators (see [1,2,3,7] and [9] for references). Recall that an operator A ∈ B(H) is said to be (p, k)-quasihyponormal, A ∈ pk − QH, for some real number 0 < p ≤ 1 and non-negative integer k (momentarily, we allow…”