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Abstract
We show that a bounded linear operator is a multiple of a unitary operator if and only if
and
always have the same numerical radius or the same numerical range for all (rank one)
. More generally, for any bounded linear operators
, we show that
and
always have the same numerical radius (resp., the same numerical range) for all (rank one)
if and only if
(resp.,
) is a multiple of a unitary operator for some
. We extend the result to other types of generalized numerical ranges including the
-numerical range and the higher rank numerical range.
Acknowledgements
The authors would like to thank Ngai-Ching Wong for some stimulating discussion.
Disclosure statement
No potential conflict of interest was reported by the authors.
Additional information
Funding
The research of Gau, Tsai and Wang was partially supported by the Ministry of Science and Technology of the Republic of China under projects MOST 103-2115-M-008-006, MOST 103-2811-M-110-027 and MOST 103-2115-M-009-001, respectively. The research of Li was partially supported by NSF USA; he is an affiliate member of the Institute for Quantum Computing at University of Waterloo, an honorary professor of the University of Hong Kong, and the Shanghai University. This research began when he was visiting Taiwan in the summer of 2014 supported by the Ministry of Science and Technology of the Republic of China.