Algebraic and geometric aspects of rational Γ\Gamma-inner functions

The set $$\Gamma {\stackrel{\rm def}{=}} \{(z+w,zw):|z|\leq 1,|w|\leq 1\} \subset {\mathbb{C}}^2$$ has intriguing complex-geometric properties; it has a 3-parameter group of automorphisms, its distinguished boundary is a ruled surface homeomorphic to the M\ obius band and it has a special subvariety which is the only complex geodesic that is invariant under all automorphisms. We exploit this geometry to develop an explicit and detailed structure theory for the rational maps from the unit disc to $\Gamma$ that map the boundary of the disc to the distinguished boundary of $\Gamma$. The talk is based on joint work with Jim Agler and Nicholas Young. \begin{itemize} \item[{[1]}] Jim Agler, Zinaida A. Lykova and N. J. Young: Algebraic and geometric aspects of rational $\Gamma$-inner functions, (arXiv: 1502.04216 [math.CV] 17 Febr. 2015) 22 pp. \end{itemize

Carath\'eodory extremal functions on the symmetrized bidisc

We show how realization theory can be used to find the solutions of the Carath\'eodory extremal problem on the symmetrized bidisc $$G \stackrel{\rm{def}}{=} \{(z+w,zw):|z|<1, \, |w|<1\}.$$ We show that, generically, solutions are unique up to composition with automorphisms of the disc. We also obtain formulae for large classes of extremal functions for the Carath\'eodory problems for tangents of non-generic types.Comment: 24 pages, 1 figure. This version contains some minor changes. It is to appear in a volume of Operator Theory: Advamces and Applications, Birkhause

The boundary Carath\'{e}odory-Fej\'{e}r interpolation problem

We give an elementary proof of a solvability criterion for the {\em boundary Carath\'{e}odory-Fej\'{e}r problem}: given a point $x \in \R$ and, a finite set of target values, to construct a function $f$ in the Pick class such that the first few derivatives of $f$ take on the prescribed target values at $x$. We also derive a linear fractional parametrization of the set of solutions of the interpolation problem. The proofs are based on a reduction method due to Julia and Nevanlinna.Comment: 30 pages. We have slightly improved the presentatio