Circular numerical ranges of partial isometries

Let (Formula presented.) be an (Formula presented.) -by- (Formula presented.) partial isometry whose numerical range (Formula presented.) is a circular disc with centre (Formula presented.) and radius (Formula presented.). In this paper, we are concerned with the possible values of (Formula presented.) and (Formula presented.). We show that (Formula presented.) must be (Formula presented.) if (Formula presented.) is at most (Formula presented.) and conjecture that the same is true for the general (Formula presented.). As for the radius, we show that if (Formula presented.) , then the set of all possible values of (Formula presented.) is (Formula presented.). Indeed, it is shown more precisely that for (Formula presented.) , (Formula presented.) , the possible values of (Formula presented.) are those in the interval (Formula presented.). In the proof process, we also characterize (Formula presented.) -by- (Formula presented.) 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.