On the Operator Identity ∑ A_kXB_k ≡ 0

Abstract

Let A j and B j (1 ≦ j ≦ m ) be bounded operators on a Banach space ᚕ and let Φ be the mapping on , the algebra of bounded operators on ᚕ, defined by (1) We give necessary and sufficient conditions for Φ to be identically zero or to be a compact map or (in the Hilbert space case) for the induced mapping on the Calkin algebra to be identically zero. These results are then used to obtain some results about inner derivations and, more generally, about mappings of the form For example, it is shown that the commutant of the range of C(S, T) is “small” unless S and T are scalars.