A Hankel matrix acting on Hardy and Bergman spaces

Petros Galanopoulos¹ ; José Ángel Peláez¹

¹ Departamento de Análisis Matemático Universidad de Málaga Campus de Teatinos, 29071 Málaga, Spain

Studia Mathematica, Tome 200 (2010) no. 3, pp. 201-220

Résumé

Let $\mu$ be a finite positive Borel measure on $[0,1).$ Let ${\cal H}_\mu=(\mu_{n,k})_{n,k\geq 0}$ be the Hankel matrix with entries $ \mu_{n,k}=\int_{[0,1)} t^{n+k}\,d\mu(t). $ The matrix $\mathcal{H}_{\mu}$ induces formally an operator on the space of all analytic functions in the unit disc by the fomula $$ {\cal H}_\mu(f)(z)=\sum_{n=0}^\infty\Bigl(\sum_{k= 0}^\infty\mu_{n,k}a_k\Bigr)z^n,\ \quad z\in {\mathbb D}, $$ where $f(z)=\sum_{n=0}^\infty a_n z^n$ is an analytic function in $\mathbb D$.We characterize those positive Borel measures on $[0,1)$ such that ${\cal H}_\mu(f)(z)= \int_{[0,1)}\frac{f(t)}{1-tz}\,d\mu(t)$ for all $f$ in the Hardy space $H^1$, and among them we describe those for which ${\cal H}_\mu$ is bounded and compact on $H^1$. We also study the analogous problem for the Bergman space $A^2$.

Détail

DOI : 10.4064/sm200-3-1

Keywords: finite positive borel measure cal geq hankel matrix entries int matrix mathcal induces formally operator space analytic functions unit disc fomula cal sum infty bigl sum infty k bigr quad mathbb where sum infty n analytic function mathbb characterize those positive borel measures cal int frac tz hardy space among describe those which cal bounded compact study analogous problem bergman space