On Self-commutator Approximants
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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 [수식] for some a ∈ J(X), then ||a|| ≤ ||a−(x＊x−xx＊)|| for all x ∈ J(X)∩a−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.
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 [수식] for some a ∈ J(X), then ||a|| ≤ ||a−(x＊x−xx＊)|| for all x ∈ J(X)∩a−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.
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