Łukasz Kosiński

Geometry of quasi-circular domains and applications to tetrablock

We prove that the Shilov boundary is invariant under proper holomorphic mappings between some classes of domains (containing among others quasi-balanced domains with continuous Minkowski functionals). Moreover, we obtain an extension theorem for proper holomorphic mappings between quasi-circular domains. Using these results we show that there are no non-trivial proper holomorphic self-mappings in the tetrablock. Another important result of our work is a description of Shilov boundaries of a large class of domains (containing among other the symmetrized polydisc and the tetrablock). It is also shown that the tetrablock is not -convex.

The group of automorphisms of the spectral ball

It is shown in this short note that the conjecture on the description of the group of automorphism of the spectral ball posed by Ransford and White is false.

Structure of the group of automorphisms of the spectral \(2\)-ball

We show that the group generated by triangular and diagonal conjugations is dense in

Cyclic polynomials in two variables

\mathrm Aut (\Omega _2)

The Group of Automorphisms of the Pentablock

(in the compact-uniform topology). Moreover, it is shown that any automorphism of

Proper holomorphic mappings in the special class of Reinhardt domains

\Omega _2