Integral representation formula for generalized normal derivations

For generalized normal derivations, acting on the space of all bounded Hilbert space operators, the following integral representation formulas hold:

 

and

 

whenever is a Hilbert-Schmidt class operator and is a Lipschitz class function on Applying this formula, we extend the Fuglede-Putnam theorem concerning commutativity modulo Hilbert-Schmidt class, as well as trace inequalities for covariance matrices of Muir and Wong. Some new monotone matrix functions and norm inequalities are also derived.