On certain Lie algebras of normal operators

The aim of this paper is to generalize some results concerning normal operators on Hilbert spaces. The generalizations hold for Banach space operators, in the framework introduced by M. Sabac in J. Operator Theory 31(1994), 319-326. Particularly we prove sufficient conditions for certain Lie algebras of generalized scalar operators to be abelian. We also characterize the ideally finite real Lie algebras which may be faithfully represented by normal operators on a Banach space (in the above cited framework).