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New class of operators where the distance between the identity operator and the generalized Jordan ∗-derivation range is maximal

Messaoudene Hadia, Mesbah Nadia

Demonstratio Mathematica

2021

Vol. 54, nr 1

311--317

A new class of operators, larger than ∗ -finite operators, named generalized ∗ -finite operators and noted by GF∗ (H) is introduced, where: GF∗ (H) = {(A, B) ∈ B(H) × B(H) : ∥TA - BT∗ - λI∥ ≥ ∣λ∣, ∀λ ∈ C, ∀T ∈ B(H)}. Basic properties are given. Some examples are also presented.

The Putnam-Fuglede property for paranormal and *-paranormal operators

Pagacz P.

Opuscula Mathematica

2013

Vol. 33, no. 3

565--574

An operator T ∈ B(H) is said to have the Putnam-Fuglede commutativity property (PF property for short) if T*X = XJ for any X ∈ B(K,H) and any isometry J ∈ B(K) such that TX = XJ*. The main purpose of this paper is to examine if paranormal operators have the PF property. We prove that k*-paranormal operators have the PF property. Furthermore, we give an example of a paranormal without the PF property.