A Hankel matrix acting on Hardy and Bergman spaces

Petros Galanopoulos, and José Ángel Peláez. "A Hankel matrix acting on Hardy and Bergman spaces." Studia Mathematica 200.3 (2010): 201-220. <http://eudml.org/doc/285661>.

@article{PetrosGalanopoulos2010,
abstract = {Let μ be a finite positive Borel measure on [0,1). Let $ℋ_\{μ\} = (μ_\{n,k\})_\{n,k≥0\}$ be the Hankel matrix with entries $μ_\{n,k\} = ∫_\{[0,1)\} t^\{n+k\} dμ(t)$. The matrix $_\{μ\}$ induces formally an operator on the space of all analytic functions in the unit disc by the fomula $ℋ_\{μ\}(f)(z) = ∑_\{n=0\}^\{∞\}i (∑_\{k=0\}^\{∞\} μ_\{n,k\}a_\{k\})zⁿ$, z ∈ , where $f(z) = ∑_\{n=0\}^\{∞\} aₙzⁿ$ is an analytic function in . We characterize those positive Borel measures on [0,1) such that $ℋ_\{μ\}(f)(z) = ∫_\{[0,1)\} f(t)/(1-tz) dμ(t)$ for all f in the Hardy space H¹, and among them we describe those for which $ℋ_\{μ\}$ is bounded and compact on H¹. We also study the analogous problem for the Bergman space A².},
author = {Petros Galanopoulos, José Ángel Peláez},
journal = {Studia Mathematica},
language = {eng},
number = {3},
pages = {201-220},
title = {A Hankel matrix acting on Hardy and Bergman spaces},
url = {http://eudml.org/doc/285661},
volume = {200},
year = {2010},
}

TY - JOUR
AU - Petros Galanopoulos
AU - José Ángel Peláez
TI - A Hankel matrix acting on Hardy and Bergman spaces
JO - Studia Mathematica
PY - 2010
VL - 200
IS - 3
SP - 201
EP - 220
AB - Let μ be a finite positive Borel measure on [0,1). Let $ℋ_{μ} = (μ_{n,k})_{n,k≥0}$ be the Hankel matrix with entries $μ_{n,k} = ∫_{[0,1)} t^{n+k} dμ(t)$. The matrix $_{μ}$ induces formally an operator on the space of all analytic functions in the unit disc by the fomula $ℋ_{μ}(f)(z) = ∑_{n=0}^{∞}i (∑_{k=0}^{∞} μ_{n,k}a_{k})zⁿ$, z ∈ , where $f(z) = ∑_{n=0}^{∞} aₙzⁿ$ is an analytic function in . We characterize those positive Borel measures on [0,1) such that $ℋ_{μ}(f)(z) = ∫_{[0,1)} f(t)/(1-tz) dμ(t)$ for all f in the Hardy space H¹, and among them we describe those for which $ℋ_{μ}$ is bounded and compact on H¹. We also study the analogous problem for the Bergman space A².
LA - eng
UR - http://eudml.org/doc/285661
ER -