### Boundary behaviour of analytic functions in spaces of Dirichlet type.

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Let μ be a finite positive Borel measure on [0,1). Let ${\mathscr{H}}_{\mu}={\left({\mu}_{n,k}\right)}_{n,k\ge 0}$ be the Hankel matrix with entries ${\mu}_{n,k}={\int}_{[0,1)}{t}^{n+k}d\mu \left(t\right)$. The matrix ${}_{\mu}$ induces formally an operator on the space of all analytic functions in the unit disc by the fomula ${\mathscr{H}}_{\mu}\left(f\right)\left(z\right)={\sum}_{n=0}^{\infty}i\left({\sum}_{k=0}^{\infty}{\mu}_{n,k}{a}_{k}\right)z\u207f$, z ∈ , where $f\left(z\right)={\sum}_{n=0}^{\infty}a\u2099z\u207f$ is an analytic function in . We characterize those positive Borel measures on [0,1) such that ${\mathscr{H}}_{\mu}\left(f\right)\left(z\right)={\int}_{[0,1)}f\left(t\right)/(1-tz)d\mu \left(t\right)$ for all f in the Hardy space H¹, and among them we describe those for which ${\mathscr{H}}_{\mu}$ is bounded and compact on H¹. We also study the analogous problem for the Bergman space A².

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