Inequalities for linear combinations of orthogonal projections and applications

Abstract

In this paper, we present various inequalities regarding the linear combinations oforthogonal projections. These results aim to generalize and refine well-known inequalities,such as those due to Buzano and Ostrowski. Additionally,we investigate a specificcase of these linear combinations and introduce new refinements of the Cauchy–Schwarz inequality. Furthermore, we establish some findings related to the covarianceand variance of bounded linear operators. Moreover, as applications of some of ourresults, we establish several inequalities involving the product of three operators, oneof which is a linear combination of an orthogonal projection and the identity operator.Finally, we introduce a new positive operator construction in terms of an orthogonalprojection and the identity operator, and we derive some norms and numerical radiusinequalities involving it.