A Mixed-Adjoint Framework for 2 × 2 Operator Matrices and Their Numerical Radii

We develop a semi-inner product framework for 2×2 block operator matrices acting on the direct sum K=H⊕H of a complex Hilbert space H, generated by the diagonal operator matrix A=diag(A,B), where A and B are distinct positive operators acting on H, neither assumed to be injective. This extends the setting of Bhunia, Paul, and Nayak [Electron. J. Linear Algebra 36 (2020), 143–157], who treated the case A=diag(A,A) with A injective. Using Douglas’ majorization theorem, we employ and develop the mixed (A,B)- and (B,A)-adjoint formalism in the block operator matrix setting; these adjoints relate the two distinct seminorms induced by A and B and take the place of the ordinary adjoint when these seminorms differ. Within this framework, we establish upper and lower bounds for the A-numerical radius of 2×2 operator matrices belonging to the appropriate A-bounded or A-adjointable classes, extending and refining the bounds of the above-mentioned paper to the case of two distinct, not necessarily injective, positive generators.