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

Extremal holomorphic maps and the symmetrised bidisc

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 $\mathcal{C}_\nu, \nu \geq 0,$ 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 $n\geq 3$. 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.Comment: 40 page

A Caratheodory theorem for the bidisk via Hilbert space methods

If \ph is an analytic function bounded by 1 on the bidisk \D^2 and \tau\in\tb is a point at which \ph has an angular gradient \nabla\ph(\tau) then \nabla\ph(\la) \to \nabla\ph(\tau) as \la\to\tau nontangentially in \D^2. This is an analog for the bidisk of a classical theorem of Carath\'eodory for the disk. For \ph as above, if \tau\in\tb is such that the $\liminf$ of (1-|\ph(\la)|)/(1-\|\la\|) as \la\to\tau is finite then the directional derivative D_{-\de}\ph(\tau) exists for all appropriate directions \de\in\C^2. Moreover, one can associate with \ph and $\tau$ an analytic function $h$ in the Pick class such that the value of the directional derivative can be expressed in terms of $h$

Facial behaviour of analytic functions on the bidisk

We prove that if $\phi$ is an analytic function bounded by 1 on the bidisk and $\tau$ is a point in a face of the bidisk at which $\phi$ satisfies Caratheodory's condition then both $\phi$ and the angular gradient $\nabla\phi$ exist and are constant on the face. Moreover, the class of all $\phi$ with prescribed $\phi(\tau)$ and $\nabla\phi(\tau)$ can be parametrized in terms of a function in the two-variable Pick class. As an application we solve an interpolation problem with nodes that lie on faces of the bidisk.Comment: 18 pages. We have replaced an erroneous proof of Theorem 5.4(1) by a valid proo

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

A case of mu-synthesis as a quadratic semidefinite program

We analyse a special case of the robust stabilization problem under structured uncertainty. We obtain a new criterion for the solvability of the spectral Nevanlinna-Pick problem, which is a special case of the $\mu$-synthesis problem of $H^\infty$ control in which $\mu$ is the spectral radius. Given $n$ distinct points \la_1,\dots,\la_n in the unit disc and $2\times 2$ nonscalar complex matrices $W_1,\dots,W_n$, the problem is to determine whether there is an analytic $2\times 2$ matrix function $F$ on the disc such that F(\la_j)=W_j for each $j$ and the supremum of the spectral radius of F(\la) is less than 1 for \la in the disc. The condition is that the minimum of a quadratic function of pairs of positive $3n$-square matrices subject to certain linear matrix inequalities in the data be attained and be zero.Comment: 37 pages, 4 figures. To appear in SIAM J. Control and Optimizatio

Operator monotone functions and L\"owner functions of several variables

We prove generalizations of L\"owner's results on matrix monotone functions to several variables. We give a characterization of when a function of $d$ variables is locally monotone on $d$-tuples of commuting self-adjoint $n$-by-$n$ matrices. We prove a generalization to several variables of Nevanlinna's theorem describing analytic functions that map the upper half-plane to itself and satisfy a growth condition. We use this to characterize all rational functions of two variables that are operator monotone

Nevanlinna representations in several variables

We generalize two integral representation formulae of Nevanlinna to functions of several variables. We show that for a large class of analytic functions that have non-negative imaginary part on the upper polyhalfplane there are representation formulae in terms of densely defined self-adjoint operators on a Hilbert space. We introduce three types of structured resolvent of a self-adjoint operator and identify four different types of representation in terms of these resolvents. We relate the types of representation that a function admits to its growth at infinity.Comment: 37 pages. In this version we have added some references and expanded the introductio

Geodesics, retracts, and the norm-preserving extension property in the symmetrized bidisc

A set in a domain in has the norm-preserving extension property if every bounded holomorphic function on has a holomorphic extension to with the same supremum norm. We prove that an algebraic subset of the symmetrized bidischas the norm-preserving extension property if and only if it is either a singleton, itself, a complex geodesic of , or the union of the set and a complex geodesic of degree in . We also prove that the complex geodesics in coincide with the nontrivial holomorphic retracts in . Thus, in contrast to the case of the ball or the bidisc, there are sets in which have the norm-preserving extension property but are not holomorphic retracts of . In the course of the proof we obtain a detailed classification of the complex geodesics in modulo automorphisms of . We give applications to von Neumann-type inequalities for -contractions (that is, commuting pairs of operators for which the closure of is a spectral set) and for symmetric functions of commuting pairs of contractive operators. We find three other domains that contain sets with the norm-preserving extension property which are not retracts: they are the spectral ball of matrices, the tetrablock and the pentablock. We also identify the subsets of the bidisc which have the norm-preserving extension property for symmetric functions