The commutant of the Volterra operator

It is proved by elementary methods that the strongly closed algebra generated by the Volterra operator V is the commutant of V.     

Generation and commutation properties of the Volterra operator

Let V be the classical Volterra operator on L ₂(0,1). Then the algebra generated (algebraically) by V and its adjoint is not only dense in the Banach space of all compact operators, but also in the Banach space of all Hilbert?CSchmidt operators and as well in the space ${\mathcal{B}(L_2(0,1))}$ equipped with the weak operator topology. Moreover, the algebra generated by V ² and its adjoint is dense in the Banach space of all trace class operators. We give an elementary proof that similar results are valid for polynomials in V without constant term. We also show that the commutant of any non-constant analytic function of V coincides with the commutant of V.     

Norms of powers of the Volterra operator

The norms of them-th powers of the Volterra operatorV onL ²[0, 1] are computed form=1,...,10. These numbers suggest that the sequencem!V is bounded above and below, which is shown, and that this sequence converges to 1/2, which is conjectured.     

The spectrum of a Volterra composition operator

Lyubic's conjecture, that the Volterra composition operator of equation (1) is quasinilpotent iff condition (2) holds, is established and is generalized to Volterra composition operators on L^P[0, 1].     

The Volterra operator is a compact universal quasinilpotent

Every compact quasinilpotent Hilbert space operator can be uniformly approximated by operators that are similar to the Volterra operator in L²([0,1],dx).     

On the numerical range of a Volterra operator in Lp

LetA be the linear operator inL _p (0, 1), 1<p<∞,p≠2, defined by ,x∈L _p (0, 1),s∈[0,1]. We show that the real values of numbers in the numerical range ofA have maximum , whereq=p/(p−1). This amounts to an inequality between integrals, for which we determine the case of equality.     

Asymptotic of the resolvent series of a Volterra operator and its application

The asymptotic of the resolvent kernels of some Volterra integral operators for large is studied, and its application to the question of the expansion of finite perturbations of such operators in eigen- and associated functions is given.Translated from Matematicheskie Zametki, Vol. 13, No. 6, pp. 857–868, June, 1973.     

Doubly nonlinear Volterra equations related to the p‐Laplacian operator

In this paper we consider a doubly nonlinear Volterra equation related to the p‐Laplacian with a nonsmooth kernel. By exploiting a suitable implicit time‐discretization technique we obtain the existence of global strong solution. Copyright © 2011 John Wiley & Sons, Ltd.     

Representation formula for solution of a functional equation with Volterra operator

The following functional equation is under consideration,

(0.1)     

The invariant subspaces of the shift plus integer multiple of the Volterra operator on Hardy spaces

?u?kovi? and Paudyal recently characterized the lattice of invariant subspaces of the shift plus a complex Volterra operator on the Hilbert space \(H^2\) on the unit disk. Motivated by the idea of Ong, in this paper, we give a complete characterization of the lattice of invariant subspaces of the shift operator plus a positive integer multiple of the Volterra operator on Hardy spaces \(H^p\), which essentially extends their works to the more general cases when \(1\le p<\infty \).     

Correction to: The invariant subspaces of the shift plus integer multiple of the Volterra operator on Hardy spaces

Archiv der Mathematik - We note that the main results (Proposition&nbsp;1, Proposition&nbsp;2, and Theorem&nbsp;1) in&nbsp;[1] are only true for.     

Monotonicity properties and functional inequalities for the Volterra and incomplete Volterra functions

In this paper we prove some monotonicity, log–convexity and log–concavity properties for the Volterra and incomplete Volterra functions. Moreover, as consequences of these results, we present some functional inequalities (like Turán type inequalities) as well as we determined sharp upper and lower bounds for the normalized incomplete Volterra functions in terms of weighted power means.