Abstract
Let be an
-by-
partial isometry whose numerical range
is a circular disc with centre
and radius
. In this paper, we are concerned with the possible values of
and
. We show that
must be
if
is at most
and conjecture that the same is true for the general
. As for the radius, we show that if
, then the set of all possible values of
is
. Indeed, it is shown more precisely that for
,
, the possible values of
are those in the interval
. In the proof process, we also characterize
-by-
partial isometries which are (unitarily) irreducible. The paper is concluded with a question on the rotational invariance of nilpotent partial isometries with circular numerical ranges centred at the origin.
Acknowledgements
The contents of this paper was reported by the third author in the 12th Workshop on Numerical Ranges and Numerical Radii (WORNA) at Sanya, Hainan, China on July 30, 2014. Thanks are due to the organizer C.-K. Li and the staff, especially Yanyu Fang, for all the work they have done for the workshop. We thank the (anonymous) referee for his incisive comments, which lead to improvements of our Example 4.5. The two paragraphs after it are essentially due to him.