On Symmetric Aspects of Operator Pair Inequalities in Hilbert Spaces via Pečarić’s Theorem with Applications

In this paper, we establish several new operator inequalities for generalizations of the joint numerical radius and joint operator norm for pairs of operators in complex Hilbert spaces, as well as for the classical numerical radius of a single operator. One of our main tools is the well-known Pečarić’s Theorem. As applications, we derive a series of power inequalities for the operator norm and for the generalized numerical radius, which refine and generalize a number of existing results in the literature. Our approach considers two key symmetric pairings: the Cartesian decomposition (R(U),I(U)) and the operator-adjoint pair (U,U⋆).