The norms and singular numbers of polynomials of the classical Volterra operator in L₂(0,1)

Abstract

The spectral problem (s 2 I − 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 = 0 this norm turns out to be greater than 1.2010 Mathematics Subject Classification: 47A10, 47A35, 47G10.

Volterra operator norms : a brief survey

“…After powers of V , the natural next step is to consider linear polynomials of V , namely V + µI where µ ∈ C. Using the Halmos approach, Lyubich and Tsedenbayar [16] determined the singular values of I + νV for all ν ∈ C, and thence computed the operator norm of ∥I + νV ∥ [16, Theorems 2.2 and 2.3]. Evidently ∥V + µI∥ = |µ|∥I + (1/µ)V ∥, so we can deduce the value of ∥V + µI∥.…”

“…Lyubich and Tsedenbayar conjectured in [16] that, if p(z) is any non-constant polynomial with p(0) = 1, then ∥p(V )∥ > 1. Their conjecture was later refuted by ter Elst and Zemánek [18].…”

“…The singular values of an invertible operator permit us to determine not only its norm (the largest singular value) but also the norm of its inverse (the reciprocal of the smallest singular value). Thus the computation the singular values of I + νV carried out by Lyubich and Tsedenbayar in [16] leads to a formula for the norms of (I + νV ) −1 and hence of the resolvent operators (V − µI) −1 . They did not state the formula explicitly, so we give it here.…”

“…The article [16] describes a general strategy, following the Halmos method, for computing ∥p(V )∥ for general polynomials p. However, in practice, this rapidly becomes very complicated as the degree of p increases. As already remarked in §3, even when p(z) = z d , it has only been carried out in detail for 1 ≤ d ≤ 3.…”

note on self-commutators of Volterra operator and its square

“…Various results of operator norm, the numerical range for Volterra pencils have been presented in e.g. [4], [5], [6], [7] and [8].…”