We report on the mathematical work of Paweł Domański (AMU Poznań).

We state and discuss several interrelated results, conjectures, and questions regarding contractive inequalities for classical $H^p$ spaces of the unit disc. We study both coefficient estimates in terms of weighted $\ell^2$ sums and the Riesz projection viewed as a map from $L^q$ to $H^p$ with $q\ge p$. Some numerical evidence is given that supports our conjectures.

The main purpose of this paper is the study of a~new class of summing multilinear operators acting from the product of Banach lattices with some nontrivial lattice convexity. A mixed Pietsch--Maurey--Rosenthal type factorization theorem for these operators is proved under weaker convexity requirements than the ones that are needed in the Maurey--Rosenthal factorization through products of $L^q$-spaces. A by-product of our factorization is an extension of multilinear operators defined by a~$q$-concavity type property to a product of special Banach function lattices which inherit some lattice--geometric properties of the domain spaces, as order continuity and $p$-convexity. Factorization through Fremlin's tensor products is also analyzed. Applications are presented to study a special class of linear operators between Banach function lattices that can be characterized by a strong version of $q$-concavity. This class contains $q$-dominated operators, and so the obtained results provide anew factorization theorem for operators from this class.

The notion of slicely countably determined (SCD) sets was introduced in 2010 by A. Avilés, V. Kadets, M. Martín, J. Merí and V. Shepelska. We solve in the negative some natural questions about preserving being SCD by the operations of union, intersection and Minkowski sum. Moreover, we demonstrate that corresponding examples exist in every space with the Daugavet property and can be selected to be unit balls of some equivalent norms. We also demonstrate that almost SCD sets need not be SCD, thus answering a question posed by A. Avilés et al.

We study the problem of characterizing membership of normalized holomorphic functions of the disc to the class of infinitesimal generators and their sectorial analytical extension. We provide new formulas and applications to dynamics of the corresponding semigroups using filtrations of the class of infinitesimal generators.

We study the algebra $\mathcal{E}'(\mathbb{R}^d)$ of distributions with compact support equipped with the multiplication $(T\star S)(f)=T_x(S_y(f(xy))$ where $xy=(x_1 y_1,\dots,x_d y_d)$. This allows us a very elegant access to the theory of Hadamard type operators on $C^\infty(\Omega)$, $\Omega$ open in $\mathbb{R}^d$, that is, of operators which admit all monomials as eigenvectors. We obtain a representation of the algebra of such operators as an algebra of holomorphic functions with classical Hadamard multiplication. Finally we study global solvability for such operators, in particular of Euler differential operators, on open subsets of $\mathbb{R}_+^d$.

In this note complex symmetric composition operators $C_\varphi$ on the Bergman space $A^2(\mathbb{D})$ are studied. It is shown that if an operator $C_\varphi$ is complex symmetric on $A^2(\mathbb{D})$ then either $\varphi\colon \mathbb{D}\to\mathbb{D}$ has a Denjoy--Wolff point in $\mathbb{D}$ or is an elliptic automorphism of the disc. Moreover in the latter case $\varphi$ is either a rotation or has an order smaller than six.

Let $A\left( K\right) $ be the locally convex space of all analytic germs on a compact subset $K$ of a Stein manifold $\Omega $, $\dim \Omega =n$, endowed with the standard inductive topogy, let $0^{n}$ denote the origin of $\mathbb{C}^{n}$, The main result is the characterisation of the isomorphism $A\left( K\right) \simeq A\left( \left\{ 0^{n}\right\} \right) $ in terms of pluripotential theory. It is based on the general result of Aytuna-Krone-Terzio\u{g}lu on the characterisation of power series spaces of infinite type in terms of interpolational invariants $\left( DN\right) $ and $\left( \Omega \right)$.