Abstract
In this paper we study the entropy numbers of composition operators \(S=TD\) where \(D\) is a diagonal operator generated by a sequence belonging to some generalized Lorentz sequence space and \(T\) is a linear bounded operator with image in a Banach space \(Y\). We highlight the special case of this setting where \(Y\) is a Banach space of type \(p\). Results can be used to obtain entropy estimates of absolutely convex hulls in Banach spaces of type \(p\).
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References
Aoki, T.: Locally bounded linear topological spaces. Proc. Imp. Acad. Tokyo 18, 588–594 (1942)
Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular variation. Cambridge University Press, Cambridge (1987)
Carl, B.: Entropy numbers, s-numbers, and eigenvalue problems. J. Funct. Anal. 41(3), 290–306 (1981)
Carl, B.: Entropy numbers of diagonal operators with an application to eigenvalue problems. J. Approx. Theory 32(2), 135–150 (1981)
Carl, B.: On a characterization of operators from \(l_q\) into a Banach space of type \(p\) with some applications to eigenvalue problems. J. Funct. Anal. 48(3), 394–407 (1982)
Carl, B., Edmunds, D.E.: Gelfand numbers and metric entropy of convex hulls in Hilbert spaces. Studia Math. 159(3), 391–402 (2003)
Carl, B., Hinrichs, A., Rudolph, P.: Entropy numbers of convex hulls in Banach spaces and applications (2013) to appear
Carl, B., Kyrezi, I., Pajor, A.: Metric entropy of convex hulls in Banach spaces. J. Lond. Math. Soc. (2) 60(3), 871–896 (1999)
Carl, B., Pietsch, A.: Entropy numbers of operators in Banach spaces. In: Proceeding of Fourth Prague Topological Sympos., Prague 1976, Part A, pp. 21–33, Lecture Notes in Math. 609, Springer-Verlag, Berlin (1977)
Carl, B., Stephani, I.: Entropy. Compactness and approximation of operators. Cambridge University Press, Cambridge (1990)
Diestel, J., Jarchow, H., Tonge, A.: Absolutely summing operators. Cambridge Univ. Press, Cambridge (1995)
Edmunds, D.E., Triebel, H.: Function spaces. Entropy numbers and differential operators. Cambridge Univ. Press, Cambridge (1996)
Hoffmann-Jørgensen, J.: Sums of independent Banach space valued random variables. Studia Math. 52, 159–186 (1974)
Karamata, J.: Sur un mode de croissance régulière des fonctions. Mathematica (Cluj) 4, 38–53 (1930)
Kühn, T.: Entropy numbers of diagonal operators of logarithmic type. Georgian Math. J. 8(2), 307–318 (2001)
Kühn, T.: Entropy numbers of general diagonal operators. Rev. Mat. Complut. 18(2), 479–491 (2005)
Maurey, B., Pisier, G.: Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach. (French). Studia Math. 58(1), 45–90 (1976)
Oloff, R.: Entropieeigenschaften von diagonaloperatoren. (German) Math. Nachr. 86, 157–165 (1978)
Pietsch, A.: Operator ideals. Mathematische Monographien [Mathematical Monographs] 16, VEB Deutscher Verlag der Wissenschaften, Berlin (1978)
Pietsch, A.: Eigenvalues and \(s\)-numbers. Cambridge Studies in Advanced Mathematics 13, Cambridge Univ. Press, Cambridge (1987)
Pisier, G.: Remarques sur un résultat non publié de B. Maurey. Séminaire d’Analyse Fonctionnelle, École Polytechnique, Palaiseau, Exposé 5 (1981)
Pisier, G.: The volume of convex bodies and banach space geometry. Cambridge Univ. Press, Cambridge (1989)
Rolewicz, S.: On a certain class of linear metric spaces. Bull. Acad. Polon. Sci. Cl. III(5), 471–473 (1957)
Rudolph, P.: Entropy of absolutely convex hulls. Dissertation, Friedrich-Schiller-Universität Jena (2013)
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Carl, B., Rudolph, P. Entropy numbers of operators factoring through general diagonal operators. Rev Mat Complut 27, 623–639 (2014). https://doi.org/10.1007/s13163-013-0131-5
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DOI: https://doi.org/10.1007/s13163-013-0131-5