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.