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Some results on generalized finite operators and range kernel orthogonality in Hilbert spaces

Mesbah Nadia, Messaoudene Hadia, Alharbi Asma

Demonstratio Mathematica

2021

Vol. 54, nr 1

318--325

Let be a complex Hilbert space and B(H) denotes the algebra of all bounded linear operators acting on H. In this paper, we present some new pairs of generalized finite operators. More precisely, new pairs of operators (A, B) ∈ B(H) × B(H) satisfying: ∥ AX – XB − I∥ ≥ 1, for all X ∈ B(H). An example under which the class of such operators is not invariant under similarity orbit is given. Range kernel orthogonality of generalized derivation is also studied.

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.