×

Composition operators on Hardy-Orlicz spaces. (English) Zbl 1200.47035

Mem. Am. Math. Soc. 974, v, 74 p. (2010).
Given an analytic self-map \(\phi:\mathbb D\to \mathbb D\), the composition operator \(C_\phi\) is defined by \(C_\phi(f)(z)=f(\phi(z))\). This operator is continuous on various spaces of analytic functions (Hardy spaces, Bergman spaces, Bloch, BMOA, Dirichlet spaces, etc.) and one area of research consists in analysing properties of the symbol \(\phi\) which allow to get extra properties on the operator \(C_\phi\) such as compactness, weak compactness, order boundedness, nuclearity, being \(p\)-summing, and many others.
The monograph under review is devoted to composition operators acting on Hardy-Orlicz and Bergman-Orlicz spaces and to study several of their properties on these spaces. By an Orlicz function, the authors understand \(\Psi:[0,\infty]\to [0,\infty]\) non-decreasing continuous, strictly convex such that \(\Psi(0)=0\) and \( \lim_{x\to \infty}\Psi(x)/x=\infty\). The monograph contains a chapter devoted to establish several “growth” conditions and regularity conditions on the Orlicz function \(\Phi\) which play a role in the sequel. Those which are mainly used are the classical \(\Delta_2\), meaning \(\Psi(2x)\leq K\Psi(x)\) for \(x\) large enough, and \(\Delta^2\), meaning \(\Psi(x)^2\leq \Psi(\alpha x)\) for some \(\alpha>1\) and \(x\) large enough. As usual, they differentiate between \(L^\Psi\) and the closure of \(L^\infty\) in \(L^\Psi\) denoted by \(M^\Psi\).
Chapter 3 contains the definition of the space \(H^\Psi\) defined by either analytic functions using \(L^\Psi\) norms of the restrictions \(f_r\) being bounded, or using Poisson integrals of functions defined at the boundary and whose negative Fourier coefficient vanish. Similarly, the space \(HM^\psi\) is also considered when \(L^\Psi\) is replaced by \(M^\Psi\). As in the classical spaces, they show that the Littlewood subordination principle also provides a proof of the continuity of the operator \(C_\varphi\) acting on \(H^\Psi\) spaces. They completely characterize the order bounded composition operators and show that compactness and weak compactness are equivalent in this context under the \(\Delta^2\)-condition.
An interesting example of a symbol such that \(C_\varphi\) is order bounded in \(M^\Psi\) but not \(p\)-summing for any \(p\) is provided.
Chapter 4 is devoted to Carleson measures in the setting of composition operators acting on Hardy-Orlicz spaces and its interplay with compactness. First, the authors present a result showing that two analytic functions \(\varphi_1\) and \(\varphi_2\) whose boundary values have the same modulus for which \( C_{\varphi_1}\) is compact in \(H^2\) but \( C_{\varphi_2}\) is not. They actually observe that in some cases (assuming injectivity and \(\phi_2(a)=0\) for some \(a\in \mathbb D\)) the compactness is passed from \(C_{\varphi_1}\) to \(C_{\varphi_2}\). One basic result of this chapter establishes that, if \(C_\varphi\) is compact on \(H^\Psi\) then it is also in \(H^2\), but the converse it is not true, at least under some conditions on \(\Psi\). They modify the notion of Carleson measure accordingly to the \(\Psi\)-Carleson measure and show that the corresponding “little-oh” condition characterizes compactness in this case. In the last chapter, they analyse similar problems for Bergman-Orlicz spaces.

MSC:

47B33 Linear composition operators
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] J. Arazy, S. D. Fisher, and J. Peetre, Möbius invariant function spaces, J. Reine Angew. Math. 363 (1985), 110-145. · Zbl 0566.30042 · doi:10.1007/BFb0078341
[2] Frédéric Bayart, Hardy spaces of Dirichlet series and their composition operators, Monatsh. Math. 136 (2002), no. 3, 203-236. · Zbl 1076.46017 · doi:10.1007/s00605-002-0470-7
[3] Frédéric Bayart, Compact composition operators on a Hilbert space of Dirichlet series, Illinois J. Math. 47 (2003), no. 3, 725-743. · Zbl 1059.47023
[4] Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. · Zbl 0647.46057
[5] Oscar Blasco de la Cruz and Hans Jarchow, A note on Carleson measures for Hardy spaces, Acta Sci. Math. (Szeged) 71 (2005), no. 1-2, 371-389. · Zbl 1101.47022
[6] Jun Soo Choa and Hong Oh Kim, On function-theoretic conditions characterizing compact composition operators on \(H^2\), Proc. Japan Acad. Ser. A Math. Sci. 75 (1999), no. 7, 109-112. · Zbl 0943.47021
[7] Jun Soo Choa and Hong Oh Kim, Composition operators between Nevanlinna-type spaces, J. Math. Anal. Appl. 257 (2001), no. 2, 378-402. · Zbl 0997.47022 · doi:10.1006/jmaa.2000.7356
[8] Jun Soo Choa, Hong Oh Kim, and Joel H. Shapiro, Compact composition operators on the Smirnov class, Proc. Amer. Math. Soc. 128 (2000), no. 8, 2297-2308. · Zbl 0954.47028 · doi:10.1090/S0002-9939-99-05239-9
[9] Joseph A. Cima and Alec L. Matheson, Essential norms of composition operators and Aleksandrov measures, Pacific J. Math. 179 (1997), no. 1, 59-64. · Zbl 0871.47027 · doi:10.2140/pjm.1997.179.59
[10] Carl C. Cowen and Barbara D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. · Zbl 0873.47017
[11] P. L. Duren, Theory of \(H^{p}\) spaces, Second edition, Dover Publications (2000). · Zbl 0215.20203
[12] John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. · Zbl 0469.30024
[13] Julia Gordon and Håkan Hedenmalm, The composition operators on the space of Dirichlet series with square summable coefficients, Michigan Math. J. 46 (1999), no. 2, 313-329. · Zbl 0963.47021 · doi:10.1307/mmj/1030132413
[14] Andreas Hartmann, Interpolation and harmonic majorants in big Hardy-Orlicz spaces, J. Anal. Math. 103 (2007), 197-219. · Zbl 1149.30039 · doi:10.1007/s11854-008-0006-8
[15] Herbert Hunziker and Hans Jarchow, Composition operators which improve integrability, Math. Nachr. 152 (1991), 83-99. · Zbl 0760.47015 · doi:10.1002/mana.19911520109
[16] Nizar Jaoua, Order bounded composition operators on the Hardy spaces and the Nevanlinna class, Studia Math. 134 (1999), no. 1, 35-55. · Zbl 0951.47027
[17] Nizar Jaoua, Similarity to a contraction and hypercontractivity of composition operators, Proc. Amer. Math. Soc. 129 (2001), no. 7, 2085-2092. · Zbl 0967.47017 · doi:10.1090/S0002-9939-00-05843-3
[18] Hans Jarchow, Some functional analytic properties of composition operators, Quaestiones Math. 18 (1995), no. 1-3, 229-256. First International Conference in Abstract Algebra (Kruger Park, 1993). · Zbl 0828.47026
[19] Hans Jarchow, Compactness properties of composition operators, Rend. Circ. Mat. Palermo (2) Suppl. 56 (1998), 91-97. International Workshop on Operator Theory (Cefalù, 1997). · Zbl 0934.47017
[20] Hans Jarchow and Jie Xiao, Composition operators between Nevanlinna classes and Bergman spaces with weights, J. Operator Theory 46 (2001), no. 3, suppl., 605-618. · Zbl 0996.47031
[21] Jean-Pierre Kahane, Some random series of functions, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 5, Cambridge University Press, Cambridge, 1985. · Zbl 0571.60002
[22] M. A. Krasnosel\(^{\prime}\)skiĭ and Ja. B. Rutickiĭ, Convex functions and Orlicz spaces, Translated from the first Russian edition by Leo F. Boron, P. Noordhoff Ltd., Groningen, 1961.
[23] P. Lefèvre, Some characterizations of weakly compact operators in \(H^\infty \) and on the disk algebra. Application to composition operators, J. Operator Theory 54 (2005), no. 2, 229-238. · Zbl 1104.47019
[24] P. Lefèvre, D. Li, H. Queffélec, and L. Rodríguez-Piazza, Some translation-invariant Banach function spaces which contain \(c_0\), Studia Math. 163 (2004), no. 2, 137-155. · Zbl 1048.43004 · doi:10.4064/sm163-2-3
[25] Pascal Lefevre, Daniel Li, Hervé Queffélec, and Luis Rodríguez-Piazza, A criterion of weak compactness for operators on subspaces of Orlicz spaces, J. Funct. Spaces Appl. 6 (2008), no. 3, 277-292. · Zbl 1166.46013 · doi:10.1155/2008/107568
[26] Pascal Lefèvre, Daniel Li, Hervé Queffélec, and Luis Rodríguez-Piazza, Some examples of compact composition operators on \(H^2\), J. Funct. Anal. 255 (2008), no. 11, 3098-3124. · Zbl 1157.47019 · doi:10.1016/j.jfa.2008.06.027
[27] Pascal Lefèvre, Daniel Li, Hervé Queffélec, and Luis Rodríguez-Piazza, Compact composition operators on \(H^2\) and Hardy-Orlicz spaces, J. Math. Anal. Appl. 354 (2009), no. 1, 360-371. · Zbl 1166.47027 · doi:10.1016/j.jmaa.2009.01.004
[28] Daniel Li and Hervé Queffélec, Introduction à l’étude des espaces de Banach, Cours Spécialisés [Specialized Courses], vol. 12, Société Mathématique de France, Paris, 2004 (French). Analyse et probabilités. [Analysis and probability theory]. · Zbl 1078.46001
[29] Lifang Liu, Guangfu Cao, and Xiaofeng Wang, Composition operators on Hardy-Orlicz spaces, Acta Math. Sci. Ser. B Engl. Ed. 25 (2005), no. 1, 105-111. · Zbl 1090.47016
[30] Barbara D. MacCluer, Compact composition operators on \(H^p(B_N)\), Michigan Math. J. 32 (1985), no. 2, 237-248. · Zbl 0585.47022 · doi:10.1307/mmj/1029003191
[31] Valentin Matache, A short proof of a characterization of inner functions in terms of the composition operators they induce, Rocky Mountain J. Math. 35 (2005), no. 5, 1723-1726. · Zbl 1105.47022 · doi:10.1216/rmjm/1181069659
[32] Eric A. Nordgren, Composition operators, Canad. J. Math. 20 (1968), 442-449. · Zbl 0161.34703
[33] M. M. Rao and Z. D. Ren, Theory of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 146, Marcel Dekker, Inc., New York, 1991. · Zbl 0724.46032
[34] Marvin Rosenblum and James Rovnyak, Topics in Hardy classes and univalent functions, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 1994. · Zbl 0816.30001
[35] Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. · Zbl 0925.00005
[36] Donald Sarason, Composition operators as integral operators, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp. 545-565. · Zbl 0712.47026
[37] H. J. Schwartz, Composition operators on \(H^p\), Thesis, University of Toledo (1969).
[38] Joel H. Shapiro, The essential norm of a composition operator, Ann. of Math. (2) 125 (1987), no. 2, 375-404. · Zbl 0642.47027 · doi:10.2307/1971314
[39] Joel H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. · Zbl 0791.30033
[40] Joel H. Shapiro, What do composition operators know about inner functions?, Monatsh. Math. 130 (2000), no. 1, 57-70. · Zbl 0951.47026 · doi:10.1007/s006050050087
[41] Joel H. Shapiro, Wayne Smith, and David A. Stegenga, Geometric models and compactness of composition operators, J. Funct. Anal. 127 (1995), no. 1, 21-62. · Zbl 0824.47027 · doi:10.1006/jfan.1995.1002
[42] J. H. Shapiro and P. D. Taylor, Compact, nuclear, and Hilbert-Schmidt composition operators on \(H^{2}\), Indiana Univ. Math. J. 23 (1973/74), 471-496. · Zbl 0276.47037
[43] Jan-Olov Strömberg, Bounded mean oscillation with Orlicz norms and duality of Hardy spaces, Bull. Amer. Math. Soc. 82 (1976), no. 6, 953-955. · Zbl 0351.30028 · doi:10.1090/S0002-9904-1976-14231-0
[44] Jan-Olov Strömberg, Bounded mean oscillation with Orlicz norms and duality of Hardy spaces, Indiana Univ. Math. J. 28 (1979), no. 3, 511-544. · Zbl 0429.46016 · doi:10.1512/iumj.1979.28.28037
[45] Nina Zorboska, Compact composition operators on some weighted Hardy spaces, J. Operator Theory 22 (1989), no. 2, 233-241. · Zbl 0719.47019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.