Operator versions of the Kantorovich inequality

Spain, P.

doi:10.1090/S0002-9939-96-03424-7

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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
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 $a.$

[BP] J. K. Baksalary and S. Puntanen, Generalized matrix versions of the Cauchy-Schwarz and Kantorovich inequalities, Aequationes Math. 41 (1991), no. 1, 103–110. MR 1088268 (91k:15038), http://dx.doi.org/10.1007/BF02227445
[GR] Werner Greub and Werner Rheinboldt, On a generalization of an inequality of L. V. Kantorovich, Proc. Amer. Math. Soc. 10 (1959), 407–415. MR 0105028 (21 #3774), http://dx.doi.org/10.1090/S0002-9939-1959-0105028-3
[K] L. V. Kantorovič, Functional analysis and applied mathematics, Uspehi Matem. Nauk (N.S.) 3 (1948), no. 6(28), 89–185 (Russian). MR 0027947 (10,380a)
[S] G. Strang, On the Kantorovich inequality, Proc. Amer. Math. Soc. 11 (1960), 468. MR 0112046 (22 #2904), http://dx.doi.org/10.1090/S0002-9939-1960-0112046-6

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