# A Hankel matrix acting on Hardy and Bergman spaces

Petros Galanopoulos; José Ángel Peláez

Studia Mathematica (2010)

- Volume: 200, Issue: 3, page 201-220
- ISSN: 0039-3223

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topPetros 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 -

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