Sur lʹimage et le noyau dʹune dérivation généralisée Original Research Article

Let A set membership, variant L(H1), B set membership, variant L(H2) (where H1, H2 are Hilbert spaces), and let δA, B denote the operator on L(H2, H1) given by δA, B(X)=AX−XB, XεL(H2, H1) J. P. Williams asked: For which A is R(δA)t- ∩ {A*} = {0}? (where δA, A = δA) We obtain some operators in this class. The case of δA, B, A¦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 δA. In this note we describe some classes of operators A, B for which we have R(δA, B − ∩ ker δA*, B* = {0}.