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Let Bs (H) be the real linear space of all self-adjoint operators on a complex Hilbert space H with dimH ≥ 3. It is proved that a continuous surjective map ϕ on Bs (H) preserves nonzero projections of Jordan products of two operators in both directions if and only if there exist a unitary or an anti-unitary operator U on H and a constant λ with λ² = 1 such that ϕ(A) = λU*AU for all A ∈ Bs (H). [ABSTRACT FROM AUTHOR]

In this paper, a new definition of majorization for C*-algebras is introduced. Sherman's inequality is extended to self-adjoint operators and positive linear maps by applying the method of premajorization used for com- paring two tuples of objects. A general result in a matrix setting is established. Special cases of the main theorem are studied. In particular, a HLPK-type inequality is derived. [ABSTRACT FROM AUTHOR]

Some Hardy-Hilbert type inequalities for Hilbert space operators are established. Several particular cases of interest are given as well. [ABSTRACT FROM AUTHOR]