Communications in Mathematical Analysis

Vector Inequalities For Two Projections in Hilbert Spaces and Applications

Silvestru Sever Dragomir

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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.

Article information

Source
Commun. Math. Anal., Volume 20, Number 2 (2017), 8-30.

Dates
First available in Project Euclid: 3 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.cma/1509674426

Mathematical Reviews number (MathSciNet)
MR3721799

Zentralblatt MATH identifier
06841183

Subjects
Primary: 46C05: Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) 26D15: Inequalities for sums, series and integrals 26D10: Inequalities involving derivatives and differential and integral operators

Keywords
Inner product spaces Schwarz’s inequality Buzano’s inequality Projection Operator norm Numerical radius

Citation

Dragomir, Silvestru Sever. Vector Inequalities For Two Projections in Hilbert Spaces and Applications. Commun. Math. Anal. 20 (2017), no. 2, 8--30. https://projecteuclid.org/euclid.cma/1509674426


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