Volume 13, Issue 1

These notes are based on a mini-course given at the ACOTCA conference 2025. The goal is to present full proofs of the first two key results regarding hypercyclic Toeplitz operators, in a way that is accessible to beginners.

We discuss a classical theorem of Khrushchev, which establishes that any set of positive Lebesgue measure on the circle supports a function with a continuous Cauchy transform.

This article introduces a collection of new open problems in the theory of the envelope of holomorphy and the schlichtness phenomenon. In particular, some of our problems focus on the class of truncated tube domains.

We present an old problem in Geometric Function Theory: characterizing the domains Ω ⊂ C ${\Omega}\subset \mathbb{C}$ for which every holomorphic map from the unit disk into Ω must belong to a specific function space X . We will provide some generalities about this problem, although the main aim is to delve into the geometric characterizations for several classical spaces, such as the Bloch space B $\mathcal{B}$ , the spaces of analytic functions of bounded mean oscillation BMOA, the Nevanlinna class N , the Smirnov class N + , the Hardy spaces H p , and the weighted Bergman spaces A α p ${A}_{\alpha }^{p}$ . This work synthesizes a number of seminal results by several authors, aiming to provide a unified introduction to this classical but active topic.

In this work, density of the algebra A ( D ) $\mathcal{A}\left(D\right)$ in H ( b ) spaces of finitely connected planar domains and the boundedness of composition operators on these function spaces are studied. Density of the algebra is considered when the defining function b is a non-extreme point of the unit ball of H ∞ ( D ). In the last part boundedness of composition operators on H ( b ) spaces is considered and as well as a generalization of the unit disk case is given, the boundedness of composition operators with generalized Blaschke symbols over finitely connected domains is characterized.

In this short paper we will discuss recent advances on the problem of characterizing the boundedness of the composition operator acting on the Bergman spaces A β 2 ( D 2 ) ${A}_{\beta }^{2}\left({\mathbb{D}}^{2}\right)$ whenever the self map Φ of the bidisc is induced by Rational Inner Functions. The problem stated here is submitted as part of the Problem List of the Young Researchers Workshop in Complex Analysis and Operator Theory “The Bench Math Session 2025”, organized in Jagiellonian University of Kraków at 10th–11th February 2025.