PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Generalized Hilbert Operators on Bergman and Dirichlet Spaces of Analytic Functions

Volume 63 / 2015

Sunanda Naik, Karabi Rajbangshi Bulletin Polish Acad. Sci. Math. 63 (2015), 227-235 MSC: 30H20, 47B38, 30A10, 47B35. DOI: 10.4064/ba8031-1-2016 Published online: 4 January 2016

Abstract

Let $f$ be an analytic function on the unit disk $\mathbb {D}$. We define a generalized Hilbert-type operator $\mathcal {H}_{a,b}$ by $$\mathcal {H}_{a,b}(f)(z)=\frac {\varGamma (a+1)}{\varGamma (b+1)}\int _{0}^{1}\frac {f(t)(1-t)^{b}}{(1-tz)^{a+1}} \,dt,$$ where $a$ and $b$ are non-negative real numbers. In particular, for $a=b=\beta ,\nobreakspace {}\mathcal {H}_{a,b}$ becomes the generalized Hilbert operator $\mathcal {H}_\beta $, and $\beta =0$ gives the classical Hilbert operator $\mathcal {H}$. In this article, we find conditions on $a$ and $b$ such that $\mathcal {H}_{a,b}$ is bounded on Dirichlet-type spaces $S^{p}$, $0 \lt p \lt 2$, and on Bergman spaces $A^{p}$, $2 \lt p \lt \infty .$ Also we find an upper bound for the norm of the operator $\mathcal {H}_{a,b}$. These generalize some results of E. Diamantopolous (2004) and S. Li (2009).

Authors

  • Sunanda NaikDepartment of Applied Sciences
    Gauhati University
    Guwahati 781-014, India
    e-mail
  • Karabi RajbangshiDepartment of Applied Sciences
    Gauhati University
    Guwahati 781-014, India
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image