International Seminar Series on Advances in Operator Theory and Inequalities #2

In this paper, by employing the positivity of certain block operator matrices, we establish a new class of mixed Schwarz-type inequalities. Our results provide a unified framework that not only extends but also refines several classical inequalities in operator theory. In particular, the obtained inequalities encompass and generalize well-known results such as the mixed Schwarz inequality of Kato, the functional inequality of Kittaneh, and Furuta’s extension involving mixed operator powers. As an application, we prove several new numerical radius inequalities, which provide improved estimates and unify existing results in this direction. These contributions highlight the versatility of block operator techniques in deriving operator inequalities that unify and extend a wide range of known results in the literature.