On Self-commutator Approximants

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