Abstract
We show that if T=H+iK is the Cartesian decomposition of T∈B(H), then for α,β∈R, supα2+β2=1‖αH+βK‖=w(T). We then apply it to prove that if A,B,X∈B(H) and 0≤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) denotes the real part of an operator T. A refinement of the triangle inequality is also shown.
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