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

• Autorzy: D. E. Edmunds , W. D. Evans , D. J. Harris
• Języki publikacji: EN

Abstrakty

EN

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 (a_{n}(T)) when 1 < α < ∞.

Kategorie tematyczne

• 47G10: Integral operators

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1997
• Tom: 124
• Numer: 1
• Strony: 59-80

Daty

wydano

1997

otrzymano

1996-03-22

poprawiono

1997-01-13

Twórcy

autor

D. E. Edmunds

• Centre for Mathematical Analysis and its Applications, University of Sussex, Falmer, Brighton BN1 9QH, U.K.,

autor

W. D. Evans

• School of Mathematics, University of Wales, Cardiff, Senghennydd Road, Cardiff CF2 4YH, U.K.,

autor

D. J. Harris

• School of Mathematics, University of Wales, Cardiff, Senghennydd Road, Cardiff CF2 4YH, U.K.

Bibliografia

• [1] C. Bennett and R. Sharpley, Interpolation of Operators, Pure Appl. Math. 129, Academic Press, New York, 1988.
• [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.
• [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.
• [5] D. E. Edmunds and V. D. Stepanov, On the singular numbers of certain Volterra integral operators, J. Funct. Anal. 134 (1995), 222-246.
• [6] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, 2nd ed., Cambridge Univ. Press, Cambridge, 1952.
• [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.
• [9] K. Nowak, Schatten ideal behavior of a generalized Hardy operator, Proc. Amer. Math. Soc. 118 (1993), 479-483.