BUZANO’S INEQUALITY HOLDS FOR ANY PROJECTION

Abstract

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.