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BUZANO’S INEQUALITY HOLDS FOR ANY PROJECTION

Part of: Inequalities Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  20 January 2016

S. S. DRAGOMIR
S. S. DRAGOMIR*
Affiliation:
Mathematics, School of Engineering and Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia School of Computer Science and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa email [email protected]
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Abstract

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We show that, in an inner product space $H$, the inequality

$$\begin{eqnarray}{\textstyle \frac{1}{2}}[\Vert x\Vert \,\Vert y\Vert +|\langle x,y\rangle |]\geq |\langle Px,y\rangle |\end{eqnarray}$$
is true for any vectors $x,y$ and a projection $P:H\rightarrow H$. Applications to norm and numerical radius inequalities of two bounded operators are given.

Keywords

inner product spacesSchwarz’s inequalityBuzano’s inequalityprojectionoperator normnumerical radius

MSC classification

Primary: 46C05: Hilbert and pre-Hilbert spaces
Secondary: 26D15: Inequalities for sums, series and integrals 26D10: Inequalities involving derivatives and differential and integral operators
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 3 , June 2016 , pp. 504 - 510
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

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