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Operator versions of the Kantorovich inequality


Author: P. G. Spain
Journal: Proc. Amer. Math. Soc. 124 (1996), 2813-2819
MSC (1991): Primary 47A63; Secondary 15A45, 65F65
DOI: https://doi.org/10.1090/S0002-9939-96-03424-7
MathSciNet review: 1328379
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Abstract: The Operator Kantorovich Inequality

\begin{displaymath}(R^2 - r^2) \, u^* (a^* a) \, u \ \le \ R^2 \, (u^* a^* u) (u^* a u) \end{displaymath}

holds for a wide class of operators $a$ on a Hilbert space $\mathcal {H}$ and all operators $u: % \mathcal {K}\to % \mathcal {H}$ for which $[a] \, u$ is a partial isometry, $[a]$ being the range projection of $a.$


References [Enhancements On Off] (What's this?)

  • [BP] JK Baksalary & S Puntanen, Generalized matrix versions of the Cauchy-Schwarz and Kantorovich inequalities, Aequationes Mathematicae 41 (1991), 103-110. MR 91k:15038
  • [GR] W Greub & W Rheinboldt, On a generalization of an inequality of L.V. Kantorovich, Proc American Math Soc 10 (1959), 407-415. MR 21:3774
  • [K] L V Kantorovich, Functional analysis and applied mathematics (Russian), Uspekhi Mat Nauk (NS) 3 (1948), 89-185. MR 10:380a
  • [S] W G Strang, On the Kantorovich Inequality, Proc American Math Soc 11 (1959) 468. MR 22:2904

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Additional Information

P. G. Spain
Affiliation: Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland
Email: pgs@maths.gla.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-96-03424-7
Keywords: Kantorovich Inequality, Cauchy-Schwarz Inequality
Received by editor(s): March 23, 1995
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society

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