Cartesian decomposition and numerical radius inequalities

We show that if T=H+iKT=H+iK is the Cartesian decomposition of T∈B(H)T∈B(H), then for α,β∈Rα,β∈R, supα2+β2=1⁡‖αH+βK‖=w(T)supα2+β2=1⁡‖αH+βK‖=w(T). We then apply it to prove that if A,B,X∈B(H)A,B,X∈B(H) and 0≤mI≤X0≤mI≤X, thenm‖Re(A)−Re(B)‖≤w(Re(A)X−XRe(B))≤12supθ∈R⁡‖(AX−XB)+eiθ(XA−BX)‖≤‖AX−XB‖+‖XA−BX‖2, where Re(T)Re(T) denotes the real part of an operator T. A refinement of the triangle inequality is also shown.