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.
Acknowledgements
The authors wish to give their thanks to the referees for their helpful comments and suggestions that make much improvement of this paper.
Disclosure statement
No potential conflict of interest was reported by the author(s).