کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4592360 | 1335094 | 2007 | 12 صفحه PDF | دانلود رایگان |
It is shown that if A,BA,B, and X are operators on a complex separable Hilbert space such that A and B are compact and positive, then the singular values of the generalized commutator AX−XBAX−XB are dominated by those of ‖X‖(A⊕B)‖X‖(A⊕B), where ‖.‖‖.‖ is the usual operator norm. Consequently, for every unitarily invariant norm ⦀.⦀⦀.⦀, we have⦀AX−XB⦀⩽‖X‖⦀A⊕B⦀.⦀AX−XB⦀⩽‖X‖⦀A⊕B⦀. It is also shown that if A and B are positive and X is compact, then⦀AX−XB⦀⩽max(‖A‖,‖B‖)⦀X⦀⦀AX−XB⦀⩽max(‖A‖,‖B‖)⦀X⦀ for every unitarily invariant norm. Moreover, if X is positive, then the singular values of the commutator AX−XAAX−XA are dominated by those of 12‖A‖(X⊕X). Consequently,⦀AX−XA⦀⩽12‖A‖⦀X⊕X⦀ for every unitarily invariant norm. For the usual operator norm, these norm inequalities hold without the compactness conditions, and in this case the first two norm inequalities are the same. Our inequalities include and improve upon earlier inequalities proved in this context, and they seem natural enough and applicable to be widely useful.
Journal: Journal of Functional Analysis - Volume 250, Issue 1, 1 September 2007, Pages 132–143