### A characterization of the algebra of holomorphic functions on simply connected domain.

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This article provided some sufficient or necessary conditions for a class of integral operators to be bounded on mixed norm spaces in the unit ball.

A constructive proof of the Beurling-Rudin theorem on the characterization of the closed ideals in the disk algebra A(𝔻) is given.

A generalization of the Aleksandrov operator is provided, in order to represent the adjoint of a weighted composition operator on ${\mathscr{H}}^{2}$ by means of an integral with respect to a measure. In particular, we show the existence of a family of measures which represents the adjoint of a weighted composition operator under fairly mild assumptions, and we discuss not only uniqueness but also the generalization of Aleksandrov–Clark measures which corresponds to the unweighted case, that is, to the adjoint of...

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.

Zero sets and uniqueness sets of the classical Dirichlet space are not completely characterized yet. We define the concept of admissible functions for the Dirichlet space and then apply them to obtain a new class of zero sets for . Then we discuss the relation between the zero sets of and those of ${}^{\infty}$.

We study the behaviour of the holomorphic sectional curvature (or Gaussian curvature) of the Bergman metric of planar annuli. The results are then utilized to construct a domain for which the curvature is divergent at one of its boundary points and moreover the upper limit of the curvature at that point is maximal possible, equal to 2, whereas the lower limit is -∞.