New Results on Boas-Bellman-Type Inequalities in Semi-Hilbert Spaces with Applications

In this article, we investigate new findings on Boas-Bellman-type inequalities in semi-Hilbert spaces. These spaces are generated by semi-inner products induced by positive and positive semidefinite operators. Our objective is to reveal significant properties of such spaces and apply these results to the field of multivariable operator theory. Specifically, we derive new inequalities that relate to the joint A-numerical radius, the joint operator A-seminorm, and the Euclidean A-seminorm of tuples of semi-Hilbert space operators. We assume that A is a nonzero positive operator. Our discoveries provide insights into the structure of semi-Hilbert spaces and have implications for a broad range of mathematical applications and beyond.