Functiones et Approximatio Commentarii Mathematici

Weighted spaces of holomorphic $2\pi$-periodic functions on the upper halfplane

Mohammad Ali Ardalani and Wolfgang Lusky

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Abstract

We consider spaces of $2\pi$-periodic holomorphic functions $f$ on the upper halfplane $G$ which are bounded by a~weighted sup-norm $\sup_{w \in G} |f(w)|v(w)$. Here $v: G \rightarrow ]0, \infty[$ is a function which depends essentially only on $Im(w)$, $w \in G$, and satisfies $ \lim_{t \rightarrow 0} v(it) =0$. We give a complete isomorphic classification of such spaces and investigate composition operators and the differentiation operator between them.

Article information

Source
Funct. Approx. Comment. Math., Volume 44, Number 2 (2011), 191-201.

Dates
First available in Project Euclid: 22 June 2011

Permanent link to this document
https://projecteuclid.org/euclid.facm/1308749123

Digital Object Identifier
doi:10.7169/facm/1308749123

Mathematical Reviews number (MathSciNet)
MR2841178

Zentralblatt MATH identifier
1242.46030

Subjects
Primary: 46E15: Banach spaces of continuous, differentiable or analytic functions 47B33: Composition operators

Keywords
weighted spaces holomorphic periodic functions halfplane differentiation operators composition operators

Citation

Ardalani, Mohammad Ali; Lusky, Wolfgang. Weighted spaces of holomorphic $2\pi$-periodic functions on the upper halfplane. Funct. Approx. Comment. Math. 44 (2011), no. 2, 191--201. doi:10.7169/facm/1308749123. https://projecteuclid.org/euclid.facm/1308749123


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References

  • K. D. Bierstedt and W. H. Summers, Biduals of weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Sec. A (1993), 70–79.
  • S. V. Bočkarev, Construction of polynomial bases in finite dimensional spaces of functions analytic on the disk, Proc. Steklov Inst. Math. 2 (1985), 55–81.
  • J. Bonet, Weighted spaces of holomorphic functions and operators between them, Seminar of Math. Analysis, Colecc. Abierta 64, Univ. Sevilla Secr. Publ., Sevilla (2003), 117–138.
  • J. Bonet, P. Domanski and M.Lindström, Essential norm and weak compactness of composition operators on weighted spaces of analytic functions, Canad. Math. Bull 42 (1999), 139–148.
  • J. Bourgain, Homogeneous polynomials on the ball and polynomial bases, Israel J. Math. 68 (1989), 327–347.
  • P. Domanski and M. Lindström, Sets of interpolation and sampling for weighted Banach spaces of holomorphic functions, Annales Polonici Math. LXXIX 3 (2002), 233–264.
  • A. Harutyunyan and W. Lusky, On the boundedness of the differentiation operator between weighted spaces of holomorphic functions, Studia Math. 184 (2008), 233–247.
  • S. Holtmanns, Operator representation and biduals of weighted function spaces, PhD thesis, Inst. for Math., Univ. of Paderborn (2000).
  • W. Lusky, On weighted spaces of harmonic and holomorphic functions, J. London Math. Soc. 51 (1995), 309–320.
  • W. Lusky, On the isomorphism classes of weighted spaces of harmonic and holomorphic functions, Studia Math. (1) 75 (2006), 19–45.
  • A. L. Shields and D. L. Williams, Bounded projections, duality and multipliers in spaces of analytic functions, Trans. Amer. Math. Soc. 162 (1971), 287–302.
  • A. L. Shields and D. L. Williams, Bounded projections and the growth of harmonic conjugates in the unit disc, Mich. Math. J. 29 (1982), 3–25.
  • M. A. Stanev, Weighted Banach spaces of holomorphic functions in the upper half plane, arXvi: math.FA/9911082 v1 (1999).
  • J. Taskinen, Compact composition operators on general weighted spaces, Houston J. Math. 27 (2001), 203–218.
  • P. Wojtaszczyk, On projections in spaces of bounded analytic functions with applications, Studia Math. 65 (1979), 147–173.