Approximation numbers of certain Volterra integral operators. (English) Zbl 0658.47049

Edmunds, D. E.; Evans, W. D.; Harris, D. J.

J. Lond. Math. Soc., II. Ser. 37, No. 3, 471-489 (1988).

This paper studies a Volterra integral operator of the form \[ Tf(x)=v(x)\int^{x}_{0}u(t)f(t)dt\quad (x\in {\mathbb{R}}^+=[0,\infty)), \] whose kernel satisfies only local integrability conditions.
The authors give necesary and sufficient conditions for the boundedness and compactness of T.
An objective of the paper is to obtain upper and lower bounds for the limit of the approximation numbers of T, which maps \(L^ p({\mathbb{R}}^+)\) to \(L^ q({\mathbb{R}}^+).\)
By way of illustration it is shown that, when \(u(x)=e^{Ax}\), \(v(x)=e^{-Bx}\) with \(B>A>0\) and \(p=q\in (1,\infty)\) then the mth approximation number of T is bounded above and below by multiples of \(m^{-1}\).

Reviewer: L.Hącia

MSC:

47Gxx	Integral, integro-differential, and pseudodifferential operators
47B06	Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
45P05	Integral operators

Keywords:

Volterra integral operator; local integrability; boundedness; compactness; upper and lower bounds for the limit of the approximation numbers