## Communications in Mathematical Analysis

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

### Vector Inequalities For Two Projections in Hilbert Spaces and Applications

#### 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