Abstract. Let V be the classical Volterra operator on L 2 (0, 1), and let z be a complex number. We prove that I − zV is power bounded if and only if Re z ≥ 0 and Im z = 0, while I − zV 2 is power bounded if and only if z = 0. The first result yieldsan improvement of [Py]. We also study some other related operator pencils.
Preliminaries. We say that an operatorWe recall the well-known formulaA generalization of this formula is the definition of the Riemann-Liouville integral operator of any fractional order α > 0,
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.
Abstract. We continue the paper [Ts] on the boundedness of polynomials in the Volterra operator. This provides new ways of constructing power-bounded operators. It seems interesting to point out that a similar procedure applies to the operators satisfying the Ritt resolvent condition: compare Theorem 5 and Theorem 9 below.
Let V denote the classical Volterra operator on L2[0,1] and let z1,z2 be arbitrary complex numbers. We investigated the operator norm of
$ z_{1}V+z_{2}V^{*} $.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.