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

D. Edmunds; W. Evans; D. Harris

Studia Mathematica (1997)

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

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topEdmunds, 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

top- [1] C. Bennett and R. Sharpley, Interpolation of Operators, Pure Appl. Math. 129, Academic Press, New York, 1988. Zbl0647.46057
- [2] M. Birman and M. Z. Solomyak, Estimates for the singular numbers of integral operators, Uspekhi Mat. Nauk 32 (1) (1977), 17-82 (in Russian); English transl.: Russian Math. Surveys 32 (1977).
- [3] D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Oxford Univ. Press, Oxford, 1987. Zbl0628.47017
- [4] D. E. Edmunds, W. D. Evans and D. J. Harris, Approximation numbers of certain Volterra integral operators, J. London Math. Soc. (2) 37 (1988), 471-489. Zbl0658.47049
- [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] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, 2nd ed., Cambridge Univ. Press, Cambridge, 1952. Zbl0047.05302
- [7] H. König, Eigenvalue Distribution of Compact Operators, Birkhäuser, Boston, 1989.
- [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] K. Nowak, Schatten ideal behavior of a generalized Hardy operator, Proc. Amer. Math. Soc. 118 (1993), 479-483. Zbl0791.47024

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