Generalized Hilbert Operators on Bergman and Dirichlet Spaces of Analytic Functions

@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 -