Some inequalities for communtators and an application to spectral variation

If θ -I is a positive semidefinite operator and A and B are either both Hermitian or both unitary, then every unitarily invariant norm of A-B is shown to be bounded by that of Aθ -θB. Some related inequalities are proved. An application leads to a generalization of the Lidskii-Wielandt inequality to matrices similar to Hermitian