The norms and singular numbers of polynomials of the classical Volterra operator in L2(0,1)
The spectral problem (sI − φ(V )∗φ(V ))f = 0 for an arbitrary complex polynomial φ of the classical Volterra operator V in L2(0, 1) is considered. An equivalent boundary value problem for a differential equation of order 2n, n = deg(φ), is constructed. In the case φ(z) = 1 + az the singular numbers are explicitly described in terms of roots of a transcendental equation, their localization and asymptotic behavior is investigated, and an explicit formula for the ‖I + aV ‖2 is given. For all a 6= 0 this norm turns out to be greater than 1.