A Hankel matrix acting on Hardy and Bergman spaces
Tom 200 / 2010
Streszczenie
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 $A^2$.
