Singular values, norms, and commutators

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• 作者：Omar Hirzallah ; Fuad Kittaneh
• 关键词：Singular value ; Unitarily invariant norm ; Commutator ; Compact operator ; Positive operator ; Self-adjoint operator ; Normal operator ; Inequality
• 刊名：Linear Algebra and its Applications
• 出版年：2010
• 出版时间：15 February 2010
• 年：2010
• 卷：432
• 期：5
• 页码：1322-1336
• 全文大小：207 K

文摘

Let and X_i, i=1,…,n, be bounded linear operators on a separable Hilbert space such that X_i is compact for i=1,…,n. It is shown that the singular values of are dominated by those of , where · is the usual operator norm. Among other applications of this inequality, we prove that if A and B are self-adjoint operators such that a₁Aa₂ and b₁Bb₂ for some real numbers and b₂, and if X is compact, then the singular values of the generalized commutator AX-XB are dominated by those of max(b₂-a₁,a₂-b₁)(XX). This inequality proves a recent conjecture concerning the singular values of commutators. Several inequalities for norms of commutators are also given.