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Range-Kernel orthogonality and elementary operators on certain Banach spaces

Bachir Ahmed, Segres Abdelkader, Sayyaf Nawal Ali, Ouarghi Khalid

Demonstratio Mathematica

2021

Vol. 54, nr 1

272--279

The characterization of the points in Cp:1≤p<∞(H) , the Von Neuman-Schatten p-classes, that are orthogonal to the range of elementary operators has been done for certain kinds of elementary operators. In this paper, we shall study this problem of characterization on an abstract reflexive, smooth and strictly convex Banach space for arbitrary operator. As an application, we consider other kinds of elementary operators defined on the spaces Cp:1≤p<∞(H), and finally, we give a counterexample to Mecheri’s result given in this context.

Structure of n-quasi left m-invertible and related classes of operators

Duggal Bhagwati Prashad, Kim In Hyun

Demonstratio Mathematica

2020

Vol. 53, nr 1

249--268

Given Hilbert space operators T, S ∈ B(H), let Δ and δ ∈ B(B (H)) denote the elementary operators ΔT,S(X) = (LT RS − I) (X) = TXS - X and δT,S(X) = (LT – RS)(X) = TX - XS. Let d = Δ or δ. Assuming T commutes with S∗, and choosing X to be the positive operator S∗nSn for some positive integer n, this paper exploits properties of elementary operators to study the structure of n-quasi [m, d]-operators dm T,S (X) = 0 to bring together, and improve upon, extant results for a number of classes of operators, such as n-quasi left m-invertible operators, n-quasi m-isometric operators, n-quasi m-self-adjoint operators and n-quasi (m, C) symmetric operators (for some conjugation C of H). It is proved that Sn is the perturbation by a nilpotent of the direct sum of an operator Sn1 = (…)n satisfying dmT1S1(I1) = 0 , T1 = (…) , with the 0 operator; if S is also left invertible, then Sn is similar to an operator B such that dmB∗,B(I) = 0. For power bounded S and T such that ST∗ - T∗S = 0 and ΔTS(S∗nSn) = 0, S is polaroid (i.e., isolated points of the spectrum are poles). The product property, and the perturbation by a commuting nilpotent property, of operators T, S satisfying dmT,S (I) = 0, given certain commutativity properties, transfers to operators satisfying S∗ndmT,S (I)Sn = 0.

Elementary operators still not elementary?

Mathieu M.

Opuscula Mathematica

2016

Vol. 36, no. 6

787--797

Properties of elementary operators, that is, finite sums of two-sided multiplications on a Banach algebra, have been studied under a vast variety of aspects by numerous authors. In this paper we review recent advances in a new direction that seems not to have been explored before: the question when an elementary operator is spectrally bounded or spectrally isometric. As with other investigations, a number of subtleties occur which show that elementary operators are still not elementary to handle.