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2017 Vector Inequalities For Two Projections in Hilbert Spaces and Applications
Silvestru Sever Dragomir
Commun. Math. Anal. 20(2): 8-30 (2017).

Abstract

In this paper we establish some vector inequalities related to Schwarz and Buzano results. We show amongst others that in an inner product space $H$ we have the inequality \begin{equation*} \frac{1}{4}\left[ \left\Vert x\right\Vert \left\Vert y\right\Vert +\left\vert \left\langle x,y\right\rangle -2\left\langle Px,y\right\rangle -2\left\langle Qx,y\right\rangle \right\vert \right] \geq \left\vert \left\langle QPx,y\right\rangle \right\vert \end{equation*} for any vectors $x,y$ and $P,Q$ two orthogonal projections on $H$. If $PQ=0$ we also have \begin{equation*} \frac{1}{2}\left[ \left\Vert x\right\Vert \left\Vert y\right\Vert +\left\vert \left\langle x,y\right\rangle \right\vert \right] \geq \left\vert \left\langle Px,y\right\rangle +\left\langle Qx,y\right\rangle \right\vert \end{equation*} for any $x,y\in H.$

Applications for norm and numerical radius inequalities of two bounded operators are given as well.

Citation

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Silvestru Sever Dragomir. "Vector Inequalities For Two Projections in Hilbert Spaces and Applications." Commun. Math. Anal. 20 (2) 8 - 30, 2017.

Information

Published: 2017
First available in Project Euclid: 3 November 2017

zbMATH: 06841183
MathSciNet: MR3721799

Subjects:
Primary: 26D10, 26D15, 46C05

Rights: Copyright © 2017 Mathematical Research Publishers

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Vol.20 • No. 2 • 2017
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