Abstract:

Let A E_Y(H~), B E_F(H.J ( w h ere Hi, H, are Hilbert spaces), and let S,, H
denote the operator on _Y( H,, H,) given by
6,,,(X) = AX - XB, X =.Y( H,, H,)
J. P. Williams asked: For which A is R(i%,)-n {A*) = {O}?(w here S,, A = 6,) We
obtain some operators in this class. The case of 6, s, A f B, is interesting in itself;
moreover it is useful if we have to use a decomposition of the Hilbert space in a direct
some for the consideration of 6,. In this note we describe some classes of operators
A, B for which we have R(6,, H) - f? ker S,,, se = (0). 0 1998 Elsevier Science