Extremal holomorphic maps and the symmetrised bidisc

Abstract

We introduce the class of $n$-extremal holomorphic maps, a class that generalises both finite Blaschke products and complex geodesics, and apply the notion to the finite interpolation problem for analytic functions from the open unit disc into the symmetrised bidisc $\Gamma$. We show that a well-known necessary condition for the solvability of such an interpolation problem is not sufficient whenever the number of interpolation nodes is 3 or greater. We introduce a sequence ${\mathcal{C}}_{\nu },\nu \ge 0,$ of necessary conditions for solvability, prove that they are of strictly increasing strength and show that ${\mathcal{C}}_{n-3}$ is insufficient for the solvability of an $n$-point problem for $n\ge 3$. We propose the conjecture that condition ${\mathcal{C}}_{n-2}$ is necessary and sufficient for the solvability of an $n$-point interpolation problem for $\Gamma$ and we explore the implications of this conjecture. We introduce a classification of rational $\Gamma$-inner functions, that is, analytic functions from the disc into $\Gamma$ whose radial limits at almost all points on the unit circle lie in the distinguished boundary of $\Gamma$. The classes are related to $n$-extremality and the conditions ${\mathcal{C}}_{\nu }$; we prove numerous strict inclusions between the classes.

40 pages

References in corpus (3)

• Extremal holomorphic maps and the symmetrised bidisc
• Some analysable instances of mu-synthesis
• The boundary analog of the Carathéodory-Schur interpolation problem

Cited by in corpus (15)

• Extremal holomorphic maps and the symmetrised bidisc
• Finite Blaschke products and the construction of rational $\Gamma$-inner functions
• Rigidity of Proper Holomorphic Self-mappings of the Pentablock
• Three-point Nevanlinna Pick problem in the polydisc
• (Weak) $m$-extremals and $m$-geodesics
• Coman conjecture for the bidisc
• Schwarz lemmas via the pluricomplex Green's function
• Spectral Nevanlinna-Pick problem and weak extremals in the symmetrized bidisc
• A Schwarz lemma for the symmetrized tridisc and description of interpolating functions
• Rational dilation on the symmetrized tridisc: failure, success and unknown
• Operator theory and representation of distinguished varieties in the symmetrized tridisc
• A rich structure related to the construction of analytic matrix functions
• A case of mu-synthesis as a quadratic semidefinite program
• Kähler submanifolds of the symmetrized polydisc
• The Schwarz Lemma at the Boundary of the Symmetrized Bidisc