Geometry of quasi-circular domains and applications to tetrablock

Abstract

We prove that the Shilov boundary is invariant under proper holomorphic mappings between some classes of domains (containing among others quasi-balanced domains with the 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 $\mathbb{C}$-convex.

11 pages

Cited by in corpus (11)

• Operator theory on the tetrablock
• Automorphisms of normal quasi-circular domains
• Proper holomorphic mappings between symmetrized ellipsoids
• Geometric properties of domains related to $\mu$-synthesis
• The Lempert theorem and the tetrablock
• The group of automorphisms of the pentablock
• Nevanlinna-Pick problem and uniqueness of left inverses in convex domains, symmetrized bidisc and tetrablock
• On automorphisms of quasi-circular domains fixing the origin
• Spectral Nevanlinna-Pick problem and weak extremals in the symmetrized bidisc
• Geometric properties of the tetrablock
• Proper holomorphic mappings vs. peak points and Silov boundary