ON THE IMAGE AND KERNEL OF A GENERALIZED DERIVATION

Let A is an element of L(H-1), B is an element of L(H-2) (where H-1, H -2 are Hilbert spaces), and let delta(A, B) denote the operator on L(H -2, H-1) given by delta(A, B)(X) = AX - XB, X is an element of L(H-2, H-1) J. P. Williams asked: For which A is R(delta(A))(-)boolean AND{A }= {0}? (where delta(A, A) = delta(A)). We obtain some operators in th is class. The case of delta(A, B), A not equal B, is interesting in it self; moreover it is useful if we have to use a decomposition of the H ilbert space in a direct some for the consideration of delta(A). In th is note we describe some classes of operators A, B for which we have R (delta(A, B)) - boolean AND ker delta(A, B*) = {0}. (C) 1998 Elsevier Science Inc.