Generalized Hilbert Operators on Bergman and Dirichlet Spaces of Analytic Functions

Sunanda Naik, and Karabi Rajbangshi. "Generalized Hilbert Operators on Bergman and Dirichlet Spaces of Analytic Functions." Bulletin of the Polish Academy of Sciences. Mathematics 63.3 (2015): 227-235. <http://eudml.org/doc/281137>.

@article{SunandaNaik2015,
abstract = {Let f be an analytic function on the unit disk . We define a generalized Hilbert-type operator $_\{a,b\}$ by $_\{a,b\}(f)(z) = Γ(a+1)/Γ(b+1) ∫_\{0\}^\{1\} (f(t)(1-t)^\{b\})/((1-tz)^\{a+1\}) dt$, where a and b are non-negative real numbers. In particular, for a = b = β, $_\{a,b\}$ becomes the generalized Hilbert operator $_β$, and β = 0 gives the classical Hilbert operator . In this article, we find conditions on a and b such that $_\{a,b\}$ 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 $_\{a,b\}$. These generalize some results of E. Diamantopolous (2004) and S. Li (2009).},
author = {Sunanda Naik, Karabi Rajbangshi},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {generalized Hilbert operators; integral operators; Bergman spaces; Dirichlet spaces},
language = {eng},
number = {3},
pages = {227-235},
title = {Generalized Hilbert Operators on Bergman and Dirichlet Spaces of Analytic Functions},
url = {http://eudml.org/doc/281137},
volume = {63},
year = {2015},
}

TY - JOUR
AU - Sunanda Naik
AU - Karabi Rajbangshi
TI - Generalized Hilbert Operators on Bergman and Dirichlet Spaces of Analytic Functions
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2015
VL - 63
IS - 3
SP - 227
EP - 235
AB - Let f be an analytic function on the unit disk . We define a generalized Hilbert-type operator $_{a,b}$ by $_{a,b}(f)(z) = Γ(a+1)/Γ(b+1) ∫_{0}^{1} (f(t)(1-t)^{b})/((1-tz)^{a+1}) dt$, where a and b are non-negative real numbers. In particular, for a = b = β, $_{a,b}$ becomes the generalized Hilbert operator $_β$, and β = 0 gives the classical Hilbert operator . In this article, we find conditions on a and b such that $_{a,b}$ 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 $_{a,b}$. These generalize some results of E. Diamantopolous (2004) and S. Li (2009).
LA - eng
KW - generalized Hilbert operators; integral operators; Bergman spaces; Dirichlet spaces
UR - http://eudml.org/doc/281137
ER -