A-Joint Numerical Radius Bounds for Operator Pairs in Semi-Hilbert Spaces with Applications

Consider a complex Hilbert space X,⟨·,·⟩ equipped with a positive (semidefinite) bounded linear operator A on X. The A-joint numerical radius for two A-bounded operators T and S is defined as ωA,e(T,S)=sup∥x∥A=1⟨Tx,x⟩A2+⟨Sx,x⟩A2. Among other results, in this study we demonstrate that ωe,A2T,S≤21−1rmaxTA2r,SA2r+ωArS♯AT1r for r≥1 and that ωe,A2T,S≥12ωAT2+S2+12maxωAS+T,ωAS−TωAS−ωAT. Additionally, we provide several inequalities related to the A-numerical radius and A-seminorm.