# Two-sided estimates of the approximation numbers of certain Volterra integral operators

Studia Mathematica (1997)

• Volume: 124, Issue: 1, page 59-80
• ISSN: 0039-3223

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## Abstract

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We consider the Volterra integral operator $T:{L}^{p}\left({ℝ}^{+}\right)\to {L}^{p}\left({ℝ}^{+}\right)$ defined by $\left(Tf\right)\left(x\right)=v\left(x\right){ʃ}_{0}^{x}u\left(t\right)f\left(t\right)dt$. Under suitable conditions on u and v, upper and lower estimates for the approximation numbers ${a}_{n}\left(T\right)$ of T are established when 1 < p < ∞. When p = 2 these yield $li{m}_{n\to \infty }n{a}_{n}\left(T\right)={\pi }^{-1}{ʃ}_{0}^{\infty }|u\left(t\right)v\left(t\right)|dt$. We also provide upper and lower estimates for the ${\ell }^{\alpha }$ and weak ${\ell }^{\alpha }$ norms of (an(T)) when 1 < α < ∞.

## How to cite

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Edmunds, D., Evans, W., and Harris, D.. "Two-sided estimates of the approximation numbers of certain Volterra integral operators." Studia Mathematica 124.1 (1997): 59-80. <http://eudml.org/doc/216397>.

@article{Edmunds1997,
abstract = {We consider the Volterra integral operator $T:L^\{p\}(ℝ^\{+\}) → L^\{p\}(ℝ^\{+\})$ defined by $(Tf)(x) = v(x)ʃ_\{0\}^\{x\} u(t)f(t)dt$. Under suitable conditions on u and v, upper and lower estimates for the approximation numbers $a_n(T)$ of T are established when 1 < p < ∞. When p = 2 these yield $lim_\{n→∞\} na_\{n\}(T) = π^\{-1\} ʃ_\{0\}^\{∞\} |u(t)v(t)|dt$. We also provide upper and lower estimates for the $ℓ^\{α\}$ and weak $ℓ^\{α\}$ norms of (an(T)) when 1 < α < ∞.},
author = {Edmunds, D., Evans, W., Harris, D.},
journal = {Studia Mathematica},
keywords = {Volterra integral operator; estimates for the approximation number},
language = {eng},
number = {1},
pages = {59-80},
title = {Two-sided estimates of the approximation numbers of certain Volterra integral operators},
url = {http://eudml.org/doc/216397},
volume = {124},
year = {1997},
}

TY - JOUR
AU - Edmunds, D.
AU - Evans, W.
AU - Harris, D.
TI - Two-sided estimates of the approximation numbers of certain Volterra integral operators
JO - Studia Mathematica
PY - 1997
VL - 124
IS - 1
SP - 59
EP - 80
AB - We consider the Volterra integral operator $T:L^{p}(ℝ^{+}) → L^{p}(ℝ^{+})$ defined by $(Tf)(x) = v(x)ʃ_{0}^{x} u(t)f(t)dt$. Under suitable conditions on u and v, upper and lower estimates for the approximation numbers $a_n(T)$ of T are established when 1 < p < ∞. When p = 2 these yield $lim_{n→∞} na_{n}(T) = π^{-1} ʃ_{0}^{∞} |u(t)v(t)|dt$. We also provide upper and lower estimates for the $ℓ^{α}$ and weak $ℓ^{α}$ norms of (an(T)) when 1 < α < ∞.
LA - eng
KW - Volterra integral operator; estimates for the approximation number
UR - http://eudml.org/doc/216397
ER -

## References

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5. [5] D. E. Edmunds and V. D. Stepanov, On the singular numbers of certain Volterra integral operators, J. Funct. Anal. 134 (1995), 222-246. Zbl0840.45013
6. [6] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, 2nd ed., Cambridge Univ. Press, Cambridge, 1952. Zbl0047.05302
7. [7] H. König, Eigenvalue Distribution of Compact Operators, Birkhäuser, Boston, 1989.
8. [8] J. Newman and M. Solomyak, Two-sided estimates of singular values for a class of integral operators on the semi-axis, Integral Equations Operator Theory 20 (1994), 335-349. Zbl0817.47024
9. [9] K. Nowak, Schatten ideal behavior of a generalized Hardy operator, Proc. Amer. Math. Soc. 118 (1993), 479-483. Zbl0791.47024

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