On Self-commutator Approximants

• Journal title : Kyungpook mathematical journal
• Volume 49, Issue 1,  2009, pp.1-6
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2009.49.1.001
Title & Authors
On Self-commutator Approximants

Abstract
Let B(X) denote the algebra of operators on a complex Banach space X, H(X) = {h $\small{{\in}}$ B(X) : h is hermitian}, and J(X) = {x $\small{{\in}}$ B(X) : x = $\small{x_1}$ + $\small{ix_2}$, $\small{x_1}$ and $\small{x_2}$ $\small{{\in}}$ H(X)}. Let $\small{{\delta}_a}$ $\small{{\in}}$ B(B(X)) denote the derivation $\small{{\delta}_a}$ = ax - xa. If J(X) is an algebra and $\small{{\delta}_a^{-1}(0){\subseteq}{\delta}_{a^*}^{-1}(0)}$ for some $\small{a{\in}J(X)}$, then $\small{{\parallel}a{\parallel}{\leq}{\parallel}a-(x^*x-xx^*){\parallel}}$ for all $\small{x{\in}J(X){\cap}{\delta}_a^{-1}(0)}$. The cases J(X) = B(H), the algebra of operators on a complex Hilbert space, and J(X) = $\small{C_p}$, the von Neumann-Schatten p-class, are considered.
Keywords
Banach space;von Neumann-Schatten p-class;derivation;kernel-range orthogonality;self-commutator;
Language
English
Cited by
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