Generalized Buzano Inequality

TL;DR

This paper generalizes the classical Buzano inequality to broader classes of bounded linear operators on inner product spaces, and introduces new inequalities related to operator norm and numerical radius.

Contribution

It extends the Buzano inequality to certain families of operators and establishes novel inequalities involving operator norm and numerical radius.

Findings

Generalized Buzano inequalities for specific operator classes

New bounds involving operator norm and numerical radius

Extensions of classical inequalities to broader operator families

Abstract

If $P$ is an orthogonal projection defined on an inner product space $\mathcal{H}$, then the inequality $$ |\langle Px, y\rangle|\leq \frac12 [\|x\|\|y\|+|\langle x, y\rangle|] $$ fulfills for any $x,y \in \mathcal{H}$ (see \cite{Dra16}). 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 $\mathcal{H}$. In addition, several new inequalities involving the norm and numerical radius of an operator are established.