Abstract
This paper settles a long-standing question by showing that in certain circumstances the entropy numbers of a map do not behave well under real interpolation. To do this, lemmas of combinatorial type are established and used to obtain lower bounds for the entropy numbers of a particular diagonal map acting between Lorentz sequence spaces. These lower bounds contradict the estimates from above that would be obtained if the behaviour of entropy numbers under real interpolation was as good as conjectured. The paper also provides sharp two-sided estimates of the entropy number e n (T) of diagonal operators \({T:l_{p}\rightarrow l_{q}, T(\left( a_{k}\right)_{k=1}^{\infty}) = (( \lambda_{k}a_{k}) _{k=1}^{\infty}) ,}\) where 0 < p < q ≤ ∞ and \({\{\lambda _{i}\}_{i=1}^{\infty}}\) is a non-increasing sequence of non-negative numbers with λ i = λ n for all i ≤ n.
Similar content being viewed by others
References
Aoki T.: Locally bounded linear topological spaces. Proc. Imp. Acad. Tokyo 18, 588–594 (1942)
Carl B.: Entropy numbers of diagonal operators with an application to eigenvalue problems. J. Approx. Theory 32, 135–150 (1981)
Cobos, F.: Interpolation theory and compactness. In: Spring School in Analysis, “Function spaces, inequalities and interpolation”, Paseky 2009, pp. 31–75. Matfyzpress, Czech Republic (2009)
Cobos F., Kühn T., Schonbek T.: One-sided compactness results for Aronszajn-Gagliardo functors. J. Funct. Anal. 106, 274–313 (1992)
Cobos F., Fernández-Martínez P., Martínez A.: Interpolation of the measure of non-compactness by the real method. Studia Math. 135, 25–38 (1999)
Cwickel M.: Real and complex interpolation and extrapolation of compact operators. Duke. Math. J. 65, 333–343 (1992)
Edmunds, D.E.: Recent developments concerning entropy and approximation numbers. In: Nonlinear Analysis, Function Spaces and Applications, vol. 5, pp. 33–76. Prometheus Publishing House, Prague (1994)
Edmunds D.E., Netrusov Yu.: Entropy numbers of embeddings of Sobolev spaces in Zygmund spaces. Studia Math. 128, 71–102 (1998)
Edmunds D.E., Triebel H.: Function spaces, entropy numbers, differential operators. Cambridge Tracts in Mathematics, vol. 120. Cambridge University Press, Cambridge (1996)
Kühn T.: A lower estimate for entropy numbers. J. Approx. Theory 110, 120–124 (2001)
Kühn, T.: Compact embeddings of function spaces. In: Proc. Intern. Symp. on Banach and Function spaces, Kitakyushu, Japan 2003, pp. 59–81. Yokohama Publishers, Yokohama (2004)
Kühn T.: Entropy numbers of general diagonal operators. Rev. Math. Complut. 18, 479–491 (2005)
Pietsch A.: Operator ideals. North-Holland, Amsterdam (1980)
Pietsch A.: History of Banach spaces and linear operators. Birkhäuser, Boston-Basel-Berlin (2007)
Rolewicz S.: On a certain class of linear metric spaces. Bull. Acad. Polon. Sci. Cl. III 5, 471–473 (1957)
Schütt C.: Entropy numbers of diagonal operators between symmetric Banach spaces. J. Approx. Theory 40, 121–128 (1984)
Vitushkin, A.G., Henkin, G. M.: Linear superposition of functions. Uspekhi Mat. Nauk 22, 77–124 (1967) (in Russian); English transl.: Russian Math. Surveys 22, 77–125 (1967)