Linear preservers of higher rank numerical ranges and radii

Abstract

Structural theorems regarding linear preservers of the higher rank numerical ranges are proved for the real linear space of bounded self-adjoint operators or the complex linear space of bounded linear operators acting on a Hilbert space. It is shown that the linear preservers of rank k-numerical ranges must be of the standard form: unitary similarity or unitary similarity followed by transposition with respect to a fixed orthonormal basis. Furthermore, it is shown that a linear preserver of the rank k-numerical radius must be a unimodular scalar multiple of a linear preserver of the rank k-numerical range.

Keywords:

• linear preservers
• higher rank numerical ranges
• bounded operators
• selfadjoint operators
• unitary operators

AMS Subject Classifications:

• 15A60
• 47A12

Acknowledgements

We thank Professor Man-Duen Choi for his stimulating comments at the numerical range workshop leading to the study of the low dimensional case. We also thank Professor Nung-Sing Sze for his short proof of Lemma 3.5. Research of S. Clark, C.-K. Li and J. Mahle was supported by the NSF CSUMS Grant DMS 0703532. Research of C.-K. Li was also supported by the NSF Grant DMS 0600859. Research of L. Rodman was supported in part by the NSF Grant DMS 0456625.