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• # Artykuł - szczegóły

## Bulletin of the Polish Academy of Sciences. Mathematics

2015 | 63 | 3 | 227-235

## Generalized Hilbert Operators on Bergman and Dirichlet Spaces of Analytic Functions

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).

227-235

wydano
2015

### Twórcy

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