The sum of unitary similarity orbits containing only special operators

Let B(H) be the algebra of bounded linear operator acting on a Hilbert space H (over the complex or real field). Characterization is given to A1,…,Ak∈B(H) such that for any unitary operators is always in a special class S of operators such as normal operators, self-adjoint operators, unitary operators. As corollaries, characterizations are given to A∈B(H) such that complex, real or nonnegative linear combinations of operators in its unitary orbit U(A)={U∗AU:Uunitary} always lie in S.