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

ABSTRACT

Let H be a separable complex Hilbert space with dim H ≥ 3, ${\mathcal{B}}_s\left(H\right)$ be the Lie algebra of all bounded self-adjoint operators on H, and let $F:i{\mathcal{B}}_s\left(H\right)\to [d,\mathrm{\infty }]$ with $d\ge 0$ be a radial unitary similarity invariant function. In this paper, a structure feature is obtained for maps φ on ${\mathcal{B}}_s\left(H\right)$ satisfying $F\left(\phi \right(A\left)\phi \right(B)-\phi (B\left)\phi \right(A\left)\right)=F(AB-BA)$ for all $A,B\in {\mathcal{B}}_s\left(H\right).$ As applications, we show that, for a surjective map φ on ${\mathcal{B}}_s\left(H\right)$, the following conditions are equivalent: φ preserves the p-norm for some $1\le p<\mathrm{\infty }$ 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 $h:{\mathcal{B}}_s\left(H\right)\to \{1,-1\}$ and a functional $g:{\mathcal{B}}_s\left(H\right)\to \mathbb{R}$ such that $\phi \left(T\right)=h\left(T\right)UT{U}^{\ast }+g\left(T\right)I$ for all $T\in {\mathcal{B}}_s\left(H\right)$. 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 $\dim H=2$ is also discussed.

Keywords:

• Self-adjoint operators
• lie product
• Schatten p-norm
• numerical range
• numerical radius
• pseudo spectrum
• pseudo spectral radius

2000 Mathematical Subject Classifications:

• 47B49
• 47A10
• 47A25

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).

Additional information

Funding

This work is partially supported by the National Natural Science Foundation of China (11671294).