On Self-commutator Approximants

Abstract

Let B(X) denote the algebra of operators on a complex Banach space X , H(X) = {h ∈ B(X) : h is hermitian}, and J(X) = {x ∈ B(X) : x = x1 + ix2, x1 and x2 ∈ H(X)}. Let δa ∈ B(B(X)) denote the derivation δa(x) = ax − xa. If J(X) is an algebra and δa −1 (0) ⊆ δ −1 a * (0) for some a ∈ J(X), then ||a|| ≤ ||a−(x * x−xx *)|| for all x ∈ J(X) ∩ δa −1 (0). The cases J(X) = B(H), the algebra of operators on a complex Hilbert space, and J(X) = Cp, the von Neumann-Schatten p-class, are considered.