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Abstrakty
Let f be an analytic function on the unit disk D. We define a generalized Hilbert-type operator Ha, b by Ha, b (f)(z) = [WZÓR], where a and b are non-negative real numbers. In particular, for a=b=β, Ha, b becomes the generalized Hilbert operator Hβ, and β=0 gives the classical Hilbert operator H. In this article, we find conditions on a and b such that Ha, b is bounded on Dirichlet-type spaces Sp, 0 < p < 2, and on Bergman spaces Ap, 2 < p < ∞. Also we find an upper bound for the norm of the operator Ha, b. These generalize some results of E. Diamantopolous (2004) and S. Li (2009).
Wydawca
Rocznik
Tom
Strony
227--235
Opis fizyczny
Bibliogr. 11 poz.
Twórcy
autor
- Department of Applied Sciences, Gauhati University, Guwahati 781-014, India
autor
- Department of Applied Sciences, Gauhati University, Guwahati 781-014, India
Bibliografia
- [D] E. Diamantopolous, Hilbert matrix on Bergman spaces, Illinois J. Math. 48 (2004), 1067–1078.
- [DS] E. Diamantopolous and A. Siskakis, Composition operators and the Hilbert matrix, Studia Math. 140 (2000), 191–198.
- [DJV] M. Dostanič, M. Jevtič and D. Vukotič, Norm of the Hilbert matrix on Bergman and Hardy spaces and a theorem of Nehari type, J. Funct. Anal. 254 (2008), 2800–2815.
- [DS1] P. L. Duren and A. P. Schuster, Bergman Spaces, Math. Surveys Monogr. 100, Amer. Math. Soc., Providence, RI, 2004.
- [HLP] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, 2nd ed., Cambridge Univ. Press, 1988.
- [L] S. Li, Generalized Hilbert operator on Dirichlet-type space, Appl. Math. Comput. 214 (2009), 304–309.
- [LS] S. Li and S. Stevič, Generalized Hilbert operator and Fejér–Riesz type inequalities on the polydisc, Acta Math. Sci. Ser. B 29 (2009), 191–200.
- [S] A. Shields, Weighted shift operators and analytic function theory, in: Topics in Operator Theory, Math. Surveys 13, Amer. Math. Soc., Providence, RI, 1974, 49–128.
- [SS] S. Stevič, Hilbert operator on the polydisc, Bull. Inst. Math. Acad. Sinica 31 (2003), 135–142.
- [Z1] X. Zhu, A class of integral operators on weighted Bergman spaces with a small parameter, Indian J. Math. 50 (2008), 381–388.
- [Z2] K. Zhu, Operator Theory in Function Spaces, 2nd ed., Math. Surveys Monogr. 138, Amer. Math. Soc., Providence, RI, 2007.
Typ dokumentu
Bibliografia
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