Precise identification of the numerical range of V^n for n ≥ 3

Determine the precise description of the numerical range W(V^n) of the n-th power of the Volterra operator V on L^2[0,1], where (Vf)(x) = ∫_0^x f(t) dt, for all integers n ≥ 3.

Background

The paper studies contractivity of Möbius transforms of operators, with a focus on conditions expressed via the numerical range of T^{-1}. For the Volterra operator V on L^2[0,1], the authors determine the numerical ranges of negative powers: W(V^{-1}) is the right half-plane, and W(V^{-n}) equals the entire complex plane for n ≥ 2.

In contrast, the numerical ranges of positive powers of V are more intricate. The set W(V) has a classical, explicitly described boundary, and W(V^2) has been characterized more recently, though its description is considerably more complicated. Beyond these, the precise identification of W(V^n) for n ≥ 3 remains unresolved, motivating the explicit open problem the authors state.