On the power boundedness of certain Volterra operator pencils

Abstract

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,

Polynomials in the Volterra and Ritt operators

“…Case b < 0. It follows from Proposition 2, the power boundedness of I − aV 1/2 (a > 0, see Proposition 3) and I − tV (t > 0, [Ts,Theorem 1]…”

“…Analyzing the behaviour of these expressions as λ → 1 + , we see that the resolvent R(λ, I − aV − zV 2 ) does not satisfy the Kreiss condition on L 2 (0, 1). See also [Ts,Theorem 3].…”

“…It can be shown that M n (I − tV ) is bounded, with respect to n, for each fixed t > 0. Indeed, an argument similar to that for Proposition 3 (see [Ts,Proposition 2]) shows that the resolvent of the operator I −tV , for a fixed t > 0, remains uniformly Abel bounded on the half-line λ > 1, which is equivalent to the Cesàro boundedness of I − tV (see [MoSaZe,Theorem 3.1]). Thus, we see one more advantage of the resolvent characterizations of various geometric properties of the powers.…”

A note on the powers of Cesàro bounded operators

“…However, we can easily verify that the analogous strong statement is true. We should mention here that the first sharper asymptotic estimates on certain Volterra operator pencils were given by D. Tsedenbayar [12]. These estimates led to improvements on an earlier result of T. Pytlik [9] (see also [8]).…”

“…Here, applying the matrix construction outlined in [11], we will prove that the answer to Zemánek's question is also negative. In our counterexample, convergence holds in the strong operator topology instead of the norm topology, which can be proved by exploiting D. Tsedenbayar's earlier result [12] on the classical Volterra operator.…”