Generalized Hilbert Operators on Bergman and Dirichlet Spaces of Analytic Functions

• Autorzy: Sunanda Naik , Karabi Rajbangshi
• 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
$𝓗_{a,b}(f)(z) = Γ(a+1)/Γ(b+1) ∫_{0}^{1} (f(t)(1-t)^{b})/((1-tz)^{a+1}) dt$,
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).

Kategorie tematyczne

• 47B38: Operators on function spaces (general)
• 30H20: Bergman spaces, Fock spaces
• 30A10: Inequalities in the complex domain
• 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2015
• Tom: 63
• Numer: 3
• Strony: 227-235

Daty

wydano

2015

Twórcy

autor

Sunanda Naik

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

autor

Karabi Rajbangshi

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

Identyfikatory

DOI

10.4064/ba8031-1-2016