Circular numerical ranges of partial isometries

Abstract

Let A be an n-by-n partial isometry whose numerical range W(A) is a circular disc with centre c and radius r. In this paper, we are concerned with the possible values of c and r. We show that c must be 0 if n is at most 4 and conjecture that the same is true for the general n. As for the radius, we show that if W(A) = {z is an element of C : vertical bar z vertical bar <= r}, then the set of all possible values of r is {0, cos(pi/(n + 1))}boolean OR [cos(pi/3), cos(pi/n)]. Indeed, it is shown more precisely that for dim ker A = k, 1 <= k <= n, the possible values of r are those in the interval [cos(pi/inverted right perpendicularn/kinverted left perpendicular + 1)), cos(pi/(n - k + 2))]. In the proof process, we also characterize n-by-n 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.