Power Bounds for the Numerical Radius of the Off-Diagonal 2 × 2 Operator Matrix

Abstract

In this paper, we employ a generalization of the Boas–Bellman inequality for inner products, as developed by Mitrinović–Pečarić–Fink, to derive several upper bounds for the 2p-th power with p≥1 of the numerical radius of the off-diagonal operator matrix 0AB*0 for any bounded linear operators A and B on a complex Hilbert space H. While the general matrix is not symmetric, a special case arises when B=A*, where the matrix becomes symmetric. This symmetry plays a crucial role in the derivation of our bounds, illustrating the importance of symmetric structures in operator theory.

Keywords

• power inequalities
• operator norm
• numerical radius
• off-diagonal 2 × 2-operator matrix
• Hilbert space