Generalized derivation, SVEP, finite ascent, range closure

Farida Lombarkia ; Sabra Megri

Filomat, Tome 34 (2020) no. 10, p. 3473

Résumé

Let X be an infinite complex Banach space and consider two bounded linear operators A, B ∈ L(X). Let L A ∈ L(L(X)) and R B ∈ L(L(X)) be the left and the right multiplication operators, respectively. The generalized derivation δ A,B ∈ L(L(X)) is defined by δ A,B (X) = (L A − R B)(X) = AX − XB. In this paper we give some sufficient conditions for δ A,B to satisfy SVEP, and we prove that δ A,B − λI has finite ascent for all complex λ, for general choices of the operators A and B, without using the range kernel orthogonality. This information is applied to prove some necessary and sufficient conditions for the range of δ A,B − λI to be closed. In [18, Propostion 2.9] Duggal et al. proved that, if asc(δ A,B − λ) ≤ 1, for all complex λ, and if either (i) A * and B have SVEP or (ii) δ * A,B has SVEP, then δ A,B − λ has closed range for all complex λ if and only if A and B are algebraic operators, we prove using the spectral theory that, if asc(δ A,B − λ) ≤ 1, for all complex λ, then δ A,B − λ has closed range, for all complex λ if and only if A and B are algebraic operators, without the additional conditions (i) or (ii).

Détail

DOI : 10.2298/FIL2010473L

Classification : 47B47, 47A11, 47A10, 47A53
Keywords: Generalized derivation operator, SVEP, operators reduced by their eigenspaces, Range closure