Generalized Hilbert Operators on Bergman and Dirichlet Spaces of Analytic Functions

• Autorzy: Naik S. , Rajbangshi K.

Identyfikatory

DOI

10.4064/ba8031-1-2016

• Języki publikacji: EN

Abstrakty

EN

Let f be an analytic function on the unit disk D. We define a generalized Hilbert-type operator H_{a, b} by H_{a, b} (f)(z) = [WZÓR], where a and b are non-negative real numbers. In particular, for a=b=β, H_{a, b} becomes the generalized Hilbert operator H_β, and β=0 gives the classical Hilbert operator H. In this article, we find conditions on a and b such that H_{a, b} is bounded on Dirichlet-type spaces S^p, 0 < p < 2, and on Bergman spaces A^p, 2 < p < ∞. Also we find an upper bound for the norm of the operator H_{a, b}. These generalize some results of E. Diamantopolous (2004) and S. Li (2009).

Słowa kluczowe

EN

generalized Hilbert operators; integral operators; Bergman spaces; Dirichlet spaces

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2015
• Tom: Vol. 63, no. 3
• Strony: 227--235
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Naik S.

• Department of Applied Sciences, Gauhati University, Guwahati 781-014, India

autor

Rajbangshi K.

• 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.