The sum of unitary similarity orbits containing only special operators

Abstract

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 A 1 , … , A k ∈ B ( H ) such that for any unitary operators U 1 , … , U k , ∑ j = 1 k U j ∗ A j U j 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 : U unitary } always lie in S .