A Hankel matrix acting on Hardy and Bergman spaces

Petros Galanopoulos; José Ángel Peláez

Studia Mathematica, Tome 196 (2010), p. 201-220 / Harvested from The Polish Digital Mathematics Library

Résumé

Let μ be a finite positive Borel measure on [0,1). Let $ℋ_{μ} = {(μ_{n, k})}_{n, k \geq 0}$ be the Hankel matrix with entries $μ_{n, k} = \int_{[0, 1)} t^{n + k} d μ (t)$ . The matrix $_{μ}$ induces formally an operator on the space of all analytic functions in the unit disc by the fomula $ℋ_{μ} (f) (z) = \sum_{n = 0}^{\infty} i (\sum_{k = 0}^{\infty} μ_{n, k} a_{k}) z ⁿ$ , z ∈ , where $f (z) = \sum_{n = 0}^{\infty} a ₙ z ⁿ$ is an analytic function in . We characterize those positive Borel measures on [0,1) such that $ℋ_{μ} (f) (z) = \int_{[0, 1)} f (t) / (1 - t z) d μ (t)$ 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.

Publié le : 2010-01-01

Zbl 1206.47024

EUDML-ID : urn:eudml:doc:285661

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     title = {A Hankel matrix acting on Hardy and Bergman spaces},
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     volume = {196},
     year = {2010},
     pages = {201-220},
     zbl = {1206.47024},
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Petros Galanopoulos; José Ángel Peláez. A Hankel matrix acting on Hardy and Bergman spaces. Studia Mathematica, Tome 196 (2010) pp. 201-220. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm200-3-1/