On Self-commutator Approximants

Abstract

Let B(X) denote the algebra of operators on a complex Banach space X, H(X) = {h 2 B(X) : h is hermitian}, and = {x 2 B(X) : x = x1 + ix2,x1 and x2 2 H(X)}. Let a 2 B(B(X)) denote the derivation a(x) = ax xa. If is an algebra and a 1 (0) 1 a (0) for some a 2 J(X), then ||a|| || a (x x xx )|| for all x 2 J(X) a 1 (0). The cases = B(H), the algebra of operators on a complex Hilbert space, and = Cp, the von Neumann-Schatten p-class, are considered. Then each x 2 has a unique representation x = x1 +ix2, x1 and x2 2 H(X), and we may define a mapping x ! x from into itself by x = x1 ix2 (= (x1 +ix2) ): with the operator norm ||.|| of B(X) is a complex Banach space such that is a continuous linear involution on (3, Lemma 8, Page 50). Recall that an operator a 2 B(X) is normal if a = a1 + ia2 2 and (a1,a2) = a1a2 a2a1 = 0. We say that an operator a 2 satisfies the PF- property, short for the Putnam-Fuglede property, if a 1 (0) a 1 (0). Normal operators satisfy the PF-property: if a = a1 +ia2 is normal, then ax = 0 implies a1x = a2x = 0 =) a x = 0 (4, Page 124). Let a 2 B(B(X)) denote the derivation a(x) = ax xa = (La Ra)x, where La and Ra denote, respectively, the operators of left multiplication and right multiplication by a. If a 2 H(X), then La, Ra and La Ra 2 H(X). Evidently, if a = a1 + ia2, then a = a1 + i a2 , where ( a1 , a2 ) = 0 whenever (a1,a2) = 0.