Some refinements of the Cauchy-Schwarz inequality via orthogonal projections

Abstract

In this paper, we present several refinements of the operator Cauchy-Schwarz inequality for positive operators. Our main result strengthens the classical form of this inequality and serves as a foundation for deriving a series of new inequalities that both generalize and improve upon existing results in the literature. Furthermore, we investigate substantial improvements to the Cauchy–Schwarz inequality by employing orthogonal projections, leading to sharper bounds in various settings. Additionally, we obtain a new perspective on the Cauchy–Schwarz inequality by showing that both the inner product and the product of norms can be characterized as extremal values of projection-dependent expressions. Several related inequalities are also established, many of which recover or extend recent contributions by other authors.