A formula of A-spectral radius for A1/2-adjoint operators on semi-Hilbertian spaces

In this paper, we prove the relation rA(T) + rA(T^) + |rA(T^) - rA(T)|2 = \ |λ|: λ ∈ σA(T)\, where A is a positive semidefinite operator (not necessarily to have a closed range) and rA(T) is the A-spectral radius of T in B_A1/2(H). Also we prove that \ |λ|: λ ∈ σA(T)\ = rA(T), when T ∈ B_A1/2(H) commutes with A. By introducing A-Harte spectrum σ_Aₕ(T) of a d-tuple operator T= (T₁,,Td) ∈ (B_A1/2(H))ᵈ, we prove that r_Aₕ(T) ≤ \\|λ\|₂: λ ∈ σ_Aₕ(T)\, where r_Aₕ(T) is the A-Harte spectral radius of T.