Preservers of radial unitary similarity functions on Lie products of self-adjoint operators

Abstract

ABSTRACT Let H be a separable complex Hilbert space with dim H ≥ 3, be the Lie algebra of all bounded self-adjoint operators on H, and let with be a radial unitary similarity invariant function. In this paper, a structure feature is obtained for maps φ on satisfying for all As applications, we show that, for a surjective map φ on , the following conditions are equivalent: φ preserves the p-norm for some on Lie products; φ preserves the numerical radius on Lie products; φ preserves the pseudo-spectral radius on Lie products; there exists a unitary or conjugate unitary operator U on H, a sign function and a functional such that for all . We also show that the following conditions are equivalent: φ preserves the numerical range on Lie products; φ preserves the pseudo spectrum on Lie products. Moreover, the concrete forms of the above preservers are given. The case is also discussed.