Generalized derivation, SVEP, finite ascent, range closure

Let X be an infinite complex Banach space and consider two bounded linear operators A,B ? L(X). Let LA ? L(L(X)) and RB ? L(L(X)) be the left and the right multiplication operators, respectively. The generalized derivation ?A,B ? L(L(X)) is defined by ?A,B(X) = (LA-RB)(X) = AX-XB. In this paper we give some sufficient conditions for ?A,B to satisfy SVEP, and we prove that ?A,B-?I has finite ascent for all complex ?, 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 ?A,B-?I to be closed. In [18, Propostion 2.9] Duggal et al. proved that, if asc(?A,B-?)? 1, for all complex ?, and if either (i) A* and B have SVEP or (ii)?* A,B has SVEP, then ?A,B-? has closed range for all complex ? if and only if A and B are algebraic operators, we prove using the spectral theory that, if asc(?A,B-?) ? 1, for all complex ?, then ?A,B-? has closed range, for all complex ? if and only if A and B are algebraic operators, without the additional conditions (i) or (ii).