Banach Journal of Mathematical Analysis

A generalized Hilbert operator acting on conformally invariant spaces

Daniel Girela and Noel Merchán

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Abstract

If μ is a positive Borel measure on the interval [0,1), we let Hμ be the Hankel matrix Hμ=(μn,k)n,k0 with entries μn,k=μn+k, where, for n=0,1,2,, μn denotes the moment of order n of μ. This matrix formally induces the operator

Hμ(f)(z)=n=0(k=0μn,kak)zn on the space of all analytic functions f(z)=k=0akzk, in the unit disk D. This is a natural generalization of the classical Hilbert operator. The action of the operators Hμ on Hardy spaces has been recently studied. This article is devoted to a study of the operators Hμ acting on certain conformally invariant spaces of analytic functions on the disk such as the Bloch space, the space BMOA, the analytic Besov spaces, and the Qs-spaces.

Article information

Source
Banach J. Math. Anal., Volume 12, Number 2 (2018), 374-398.

Dates
Received: 25 December 2016
Accepted: 16 May 2017
First available in Project Euclid: 8 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1504857614

Digital Object Identifier
doi:10.1215/17358787-2017-0023

Mathematical Reviews number (MathSciNet)
MR3779719

Zentralblatt MATH identifier
06873506

Subjects
Primary: 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15]
Secondary: 30H10: Hardy spaces

Keywords
Hilbert operators conformally invariant spaces Carleson measures

Citation

Girela, Daniel; Merchán, Noel. A generalized Hilbert operator acting on conformally invariant spaces. Banach J. Math. Anal. 12 (2018), no. 2, 374--398. doi:10.1215/17358787-2017-0023. https://projecteuclid.org/euclid.bjma/1504857614


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References

  • [1] A. Aleman, A. Montes-Rodríguez, and A. Sarafoleanu, The eigenfunctions of the Hilbert matrix, Const. Approx. 36 (2012), no. 3, 353–374.
  • [2] A. Aleman and A. G. Siskakis, Integration operators on Bergman spaces, Indiana Univ. Math. J. 46 (1997), no. 2, 337–356.
  • [3] J. M. Anderson, J. G. Clunie, and C. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), 12–37.
  • [4] J. M. Anderson and A. L. Shields, Coefficient multipliers of Bloch functions, Trans. Amer. Math. Soc. 224 (1976), no. 2, 255–265.
  • [5] J. Arazy, S. D. Fisher, and J. Peetre, Möbius invariant function spaces, J. Reine Angew. Math. 363 (1985), 110–145.
  • [6] R. Aulaskari and P. Lappan, “Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal” in Complex Analysis and Its Applications (Hong Kong, 1993), Pitman Res. Notes Math. Ser. 305, Longman Sci. Tech., Harlow, 1994, 136–146.
  • [7] R. Aulaskari, P. Lappan, J. Xiao, and R. Zhao, On $\alpha$-Bloch spaces and multipliers on Dirichlet spaces, J. Math. Anal. Appl. 209 (1997), no. 1, 103–121.
  • [8] R. Aulaskari, J. Xiao, and R. Zhao, On subspaces and subsets of BMOA and UBC, Analysis 15 (1995), no. 2, 101–121.
  • [9] A. Baernstein, “Analytic functions of bounded mean oscillation” in Aspects of Contemporary Complex Analysis (Durham, 1979), edited by D. A. Brannan and J. G. Clunie, Academic Press, London, 1980, 3–36.
  • [10] G. Bao and H. Wulan, Hankel matrices acting on Dirichlet spaces, J. Math. Anal. Appl. 409 (2014), no. 1, 228–235.
  • [11] L. Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547–559.
  • [12] Ch. Chatzifountas, D. Girela, and J. Á. Peláez, A generalized Hilbert matrix acting on Hardy spaces, J. Math. Anal. Appl. 413 (2014), no. 1, 154–168.
  • [13] E. Diamantopoulos, Hilbert matrix on Bergman spaces, Illinois J. Math. 48 (2004), no. 3, 1067–1078.
  • [14] E. Diamantopoulos and A. G. Siskakis, Composition operators and the Hilbert matrix, Studia Math. 140 (2000), no. 2, 191–198.
  • [15] J. J. Donaire, D. Girela, and D. Vukotić, On univalent functions in some Möbius invariant spaces, J. Reine Angew. Math. 553 (2002), 43–72.
  • [16] M. Dostanić, M. Jevtić, and D. Vukotić, Norm of the Hilbert matrix on Bergman and Hardy spaces and a theorem of Nehari type, J. Funct. Anal. 254 (2008), no. 11, 2800–2815.
  • [17] P. L. Duren, Extension of a theorem of Carleson, Bull. Amer. Math. Soc. (N.S.) 75 (1969), 143–146.
  • [18] P. L. Duren, Theory of $H^{p}$ Spaces, Pure Appl. Math. 38, Academic Press, New York, 1970.
  • [19] P. Galanopoulos, D. Girela, J. A. Peláez, and A. G. Siskakis, Generalized Hilbert operators, Ann. Acad. Sci. Fenn. Math. 39 (2014), no. 1, 231–258.
  • [20] P. Galanopoulos and J. A. Peláez, A Hankel matrix acting on Hardy and Bergman spaces, Studia Math. 200, 3, (2010), no. 3, 201–220.
  • [21] D. Girela, “Analytic functions of bounded mean oscillation” in Complex Function Spaces (Mekrijärvi 1999), Univ. Joensuu Dept. Math. Rep. Ser. 4, Univ. Joensuu, Joensuu, 2001, 61–170.
  • [22] G. H. Hardy and J. E. Littlewood, Notes on the theory of series XIII: Some new properties of Fourier constants, J. London. Math. Soc. S1-6, (1931), no. 1, 3–9.
  • [23] F. Holland and D. Walsh, Growth estimates for functions in the Besov spaces $A_{p}$, Proc. Roy. Irish Acad. Sect. A 88 (1988), no. 1, 1–18.
  • [24] B. Lanucha, M. Nowak, and M. Pavlović, Hilbert matrix operator on spaces of analytic functions, Ann. Acad. Sci. Fenn. Math. 37 (2012), no. 1, 161–174.
  • [25] M. Mateljević and M. Pavlović, $L^{p}$-behaviour of the integral means of analytic functions, Studia Math. 77 (1984), no. 3, 219–237.
  • [26] M. Pavlović, Introduction to Function Spaces on the Disk, Posebna Izdan. 20, Matematički Institut SANU, Belgrade, 2004.
  • [27] M. Pavlović, Analytic functions with decreasing coefficients and Hardy and Bloch spaces, Proc. Edinb. Math. Soc. (2) 56 (2013), no. 2, 623–635.
  • [28] M. Pavlović, Invariant Besov spaces: Taylor coefficients and applications, preprint, http://www.researchgate.net/publication/304781567 (accessed 25July 2017).
  • [29] J. A. Peláez and J. Rättyä, Weighted Bergman spaces induced by rapidly increasing weights, Mem. Amer. Math. Soc. 227 (2014), no. 1066.
  • [30] L. E. Rubel and R. M. Timoney, An extremal property of the Bloch space, Proc. Amer. Math. Soc. 75 (1979), no. 1, 45–49.
  • [31] J. Xiao, Holomorphic $Q$ classes, Lecture Notes in Math. 1767, Springer, Berlin, 2001.
  • [32] R. Zhao, On logarithmic Carleson measures, Acta Sci. Math. (Szeged) 69 (2003), no. 3–4, 605–618.
  • [33] K. Zhu, Analytic Besov spaces, J. Math. Anal. Appl. 157 (1991), no. 2, 318–336.
  • [34] K. Zhu, Operator Theory in Function Spaces, 2nd ed., Math. Surveys Monogr. 138, Amer. Math. Soc. Providence, 2007.
  • [35] A. Zygmund, Trigonometric Series, Vols. I and II, 2nd ed., Cambridge Univ. Press, New York, 1959.