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A Hankel matrix acting on Hardy and Bergman spaces

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Galanopoulos P., Peláez J.

Studia Mathematica

2010

tom 200

nr 3

201-220

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 $𝓗_{μ}$ 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 ∈ 𝔻, where $f(z) = ∑_{n=0}^{∞} aₙzⁿ$ is an analytic function in 𝔻. 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.

Ograniczanie wyników

1 Studia Mathematica

1 Galanopoulos P.

1 Peláez J.

1 2010

Wyniki wyszukiwania

A Hankel matrix acting on Hardy and Bergman spaces