ON THE RANGE-KERNEL ORTHOGONALITY OF ELEMENTARY OPERATORS

Let L (H) denote the algebra of operators on a complex infinite dimensional Hilbert space H. For A, B is an element of L(H), the generalized derivation delta(A, B) and the elementary operator Delta (A, B) are defined by delta (A, B)(X) = A X - X B and Delta (A, B)(X) = A X B - X for all X is an element of L(H). In this paper, we exhibit pairs (A, B) of operators such that the range- kernel orthogonality of delta (A, B) holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of Delta (A, B) with respect to the wider class of unitarily invariant norms on L(H).