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Vector Inequalities For Two Projections in Hilbert Spaces and Applications

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

Vector Inequalities For Two Projections in Hilbert Spaces and Applications

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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.