The Cesàro Operator and Unconditional Taylor Series in Hardy Spaces

详细信息    查看全文

• 作者：Guillermo P. Curbera ; Werner J. Ricker
• 关键词：Primary 30H10 ; 47B37 ; 42A16 ; Secondary 47A10 ; 46B45 ; Cesàro operator ; Hardy space ; Taylor series ; unconditional convergence ; majorant property
• 刊名：Integral Equations and Operator Theory
• 出版年：2015
• 出版时间：October 2015
• 年：2015
• 卷：83
• 期：2
• 页码：179-195
• 全文大小：595 KB
• 参考文献：1.Bachelis G.F.: On the of unconditionally convergent Fourier series in L p (G). Proc. Am. Math. Soc. 27, 309-12 (1971)MATH MathSciNet
  2.Bachelis G.F., Gilbert J.E.: Banach spaces of compact multipliers and their dual spaces. Math. Z. 125, 285-97 (1972)MATH MathSciNet CrossRef
  3.Bachelis G.F.: On the upper and lower majorant properties in L p (G). Q. J. Math. Oxf. II Ser. 24, 119-28 (1973)MATH MathSciNet CrossRef
  4.Bennett G.: Factorizing the classical inequalities. Mem. Am. Math. Soc. 120(576), 1-30 (1996)
  5.Boas, R.P.: Majorant problems for trigonometric series. J. d’Anal. Math. 10, 253-71 (1962-1963)
  6.Brown A., Halmos P.R., Shields A.L.: Cesàro operators. Acta Sci. Math. (Szeged) 26, 125-37 (1965)MATH MathSciNet
  7.Curbera G.P., Ricker W.J.: Extensions of the classical Cesàro operator on Hardy spaces. Math. Scand. 108, 279-90 (2011)MATH MathSciNet
  8.Curbera G.P., Ricker W.J.: Solid extensions of the classical Cesàro operator on the Hardy space \({H^2(\mathbb{D})}\) . J. Math. Anal. Appl. 407, 387-97 (2013)MATH MathSciNet CrossRef
  9.Curbera G.P., Ricker W.J.: A feature of averaging. Integral Equ. Oper. Theory 76, 447-49 (2013)MATH MathSciNet CrossRef
  10.Curbera G.P., Ricker W.J.: Solid extensions of the Cesàro operator on \({\ell^p \,\,{\rm and}\,\, c_0}\) . Integral Equ. Oper. Theory 80, 61-7 (2014)MATH MathSciNet CrossRef
  11.Diestel J., Jarchow H., Tonge A.: Absolutely Summing Operators. Cambridge University Press, Cambridge (1995)MATH CrossRef
  12.Duren P.L.: Theory of H p Spaces. Academic Press, New York (1970)MATH
  13.Edwards R.E.: Fourier Series. A Modern Introduction, vol. I. Springer, Berlin (1982)
  14.Figà-Talamanca A.: On the subspace of L p invariant under multiplication of transform by bounded continuous functions. Rend. Semin. Mat. Univ. Padova 35, 176-89 (1965)MATH
  15.Goldberg S.: Unbounded Linear Operators: Theory and Applications. Dover, New York (1985)MATH
  16.Hardy G.H., Littlewood J.E.: Notes on the theory of series (XIX): a problem concerning majorants of Fourier series. Q. J. Math. (Oxford) 6, 304-15 (1935)CrossRef
  17.Hardy G.H., Littlewood J.E., Pólya G.: Inequalities. Cambridge University Press, Cambridge (1934)
  18.Helgason S.: Multipliers of Banach algebras. Ann. Math. 64, 240-54 (1956)MATH MathSciNet CrossRef
  19.Hobson E.W.: The Theory of Functions of a Real Variable and the Theory of Fourier’s Series, vol. II. Dover, New York (1957)
  20.Kislyakov S.V.: The Fourier coefficients of the boundary values of functions analytic in the disk and in the bidisk. Proc. Steklov Inst. Math. 155, 75-1 (1983)MATH
  21.Leibowitz G.: Spectra of discrete Cesàro operators. Tamkang J. Math. 3, 123-32 (1972)MATH MathSciNet
  22.Lindenstrauss J., Tzafriri L.: Classical Banach Spaces, vol. II. Springer, Berlin (1979)CrossRef
  23.Masty?o M., Mleczko P.: Solid hulls of quasi-Banach spaces of analytic functions and interpolation. Nonlinear Anal. 73, 84-8 (2010)MATH MathSciNet CrossRef
  24.Okada S., Ricker W.J., Rodríguez-Piazza L.: Absolutely summing convolution operators in L p (G). Proc. Lond. Math. Soc. 102, 843-82 (2011)MATH MathSciNet CrossRef
  25.Orlicz W.: über unbedingte Konvergenz in Funktionenr?umen (I). Stud. Math. 4, 33-7 (1933)
  26.Rhoades B.E.: Spectra of some Hausdorff operators. Acta Sci. Math. (Szeged) 32, 91-00 (1971)MATH MathSciNet
  27.Siskakis A.G.: Composition semigroups and the Cesàro operator on H p . J. Lond. Math. Soc. (2) 36, 153-64 (1987)MATH MathSciNet CrossRef
  28.Siskakis A.G.: The Cesàro operator is bounded on H 1. Proc. Am. Math. Soc. 110, 461-62 (1990)MATH MathSciNet
  29.Vukoti? D.: Analytic Toeplitz operators on the Hardy space H p : a survey. Bull. Belg. Math. Soc. 10, 101-13 (2003)MATH
  30.Zygmund A.: Trigomometric Series. Cambridge University Press, Cambridge (1977)
  
• 作者单位：Guillermo P. Curbera (1)
  Werner J. Ricker (2)
  
  1. Facultad de Matemáticas, IMUS, Universidad de Sevilla, Aptdo. 1160, Seville, 41080, Spain
  2. Math.-Geogr. Fakult?t, Katholische Universit?t Eichst?tt-Ingolstadt, 85072, Eichst?tt, Germany
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  
• 出版者：Birkh盲user Basel
• ISSN：1420-8989

文摘

We introduce the spaces \({H^{p}_{uc}}\) consisting of all functions in the Hardy space \({H^p, 1 < p < \infty}\), whose Taylor series are unconditionally convergent and analyze the action of the Cesàro operator in these spaces. There is a related class of Banach sequence spaces \({N^p, 1 < p < \infty}\), arising from harmonic analysis, in which the discrete Cesàro operator acts. The classical majorant property (due to Hardy and Littlewood) provides a means to transfer various results about the Cesàro operator in N p (e.g. continuity, spectrum, etc.) to those for the corresponding Cesàro operator acting in \({H^{p}_{uc}. \,\,{\rm For}\,\, p \neq 2}\), the space \({H^{p}_{uc}}\) is rather different to the classical space \({H^{p}}\). The spaces \({N^{p}}\) also exhibit a remarkable stability property under averaging, akin to that established by Bennett for \({\ell^{p}}\). Keywords Cesàro operator Hardy space Taylor series unconditional convergence majorant property