Generalized Buzano Inequality

Abstract

If P is an orthogonal projection defined on an inner product space H, then the inequality
$$
|\langle Px, y\rangle|\leq \frac12 [\|x\|\|y\|+|\langle x, y\rangle|]
$$
fulfills for any x,y \in H (see Buzano inequality holds for any projection,  S.S. Dragomir, Bull. Aust. Math. Soc. 93 (2016), no. 3, 504--510). In particular, when P is the identity operator, then it recovers the famous Buzano inequality.

We obtain generalizations of such classical inequality, which hold for certain families of bounded linear operators defined on H. In addition, several new inequalities involving the norm and numerical radius of an operator are established.