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Jim Agler

Publications and source records attributed to Jim Agler.

At least 55 records · Page 3Linked to original sources
arXiv2013

3-extremal holomorphic maps and the symmetrised bidisc

We analyse the 3-extremal holomorphic maps from the unit disc $\mathbb{D}$ to the symmetrised bidisc $ \mathcal{G}$, defined to be the set $ \{(z+w,zw): z,w\in\mathbb{D}\}$, with a view to the complex geometry and function theory of $\mathcal{G}$. These are the maps whose restriction to any triple of distinct points in $\mathbb{D}$ yields interpolation data that are only just solvable. We find a large class of such maps; they are rational of degree at most 4. It is shown that there are two qualitatively different classes of rational $\mathcal{G}$-inner functions of degree at most 4, to be called {\em aligned} and {\em caddywhompus} functions; the distinction relates to the cyclic ordering of certain associated points on the unit circle. The aligned ones are 3-extremal. We describe a method for the construction of aligned rational $\mathcal{G}$-inner functions; with the aid of this method we reduce the solution of a 3-point interpolation problem for aligned holomorphic maps from $\mathbb{D}$ to $\mathcal{G}$ to a collection of classical Nevanlinna-Pick problems with mixed interior and boundary interpolation nodes. Proofs depend on a form of duality for $\mathcal{G}$.

math.CV↗
arXiv2013

Symmetric functions of two noncommuting variables

We prove a noncommutative analogue of the fact that every symmetric analytic function of $(z,w)$ in the bidisc $\D^2$ can be expressed as an analytic function of the variables $z+w$ and $zw$. We construct an analytic nc-map $S$ from the biball to an infinite-dimensional nc-domain $Ω$ with the property that, for every bounded symmetric function $\ph$ of two noncommuting variables that is analytic on the biball, there exists a bounded analytic nc-function $Φ$ on $Ω$ such that $\ph=Φ\circ S$. We also establish a realization formula for $Φ$, and hence for $\ph$, in terms of operators on Hilbert space.

math.CV↗
arXiv2013

Global holomorphic functions in several non-commuting variables

We define a free holomorphic function to be a function that is locally a bounded nc-function. We prove that free holomorphic functions are the functions that are locally uniformly approximable by free polynomials. We prove a realization formula and an Oka-Weil theorem for free analytic functions.

math.OA↗
arXiv2013

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 $μ$-synthesis problem of $H^\infty$ control in which $μ$ 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.

math.CV↗
arXiv2012

Operator theory and the Oka extension theorem

For $δ$ an $m$-tuple of analytic functions, we define an algebra $\hidg$, contained in the bounded analytic functions on the analytic polyhedron $ {|δ^l(z)| < 1, \ 1 \leq l \leq m}$, and prove a representation formula for it. We give conditions whereby every function that is analytic on a neighborhood of $ {|δ^l(z)| \leq 1, \ 1 \leq l \leq m}$ is actually in $\hidg$. We use this to give a proof of the Oka extension theorem with bounds. We define an $\hidg$ functional calculus for operators.

math.CV↗
arXiv2012

The Takagi problem on the disk and bidisk

We give a new proof on the disk that a Pick problem can be solved by a rational function that is unimodular on the unit circle and for which the number of poles inside the disk is no more than the number of non-positive eigenvalues of the Pick matrix. We use this method to find rational solutions to Pick problems on the bidisk.

math.FA↗
arXiv2012

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.

math.CV↗
arXiv2012

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 $Γ$. 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 $\mathcal{C}_ν, ν\geq 0,$ 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 $Γ$ and we explore the implications of this conjecture. We introduce a classification of rational $Γ$-inner functions, that is, analytic functions from the disc into $Γ$ whose radial limits at almost all points on the unit circle lie in the distinguished boundary of $Γ$. The classes are related to $n$-extremality and the conditions $\mathcal{C}_ν$; we prove numerous strict inclusions between the classes.

math.CV↗
arXiv2012

Hankel vector moment sequences and the non-tangential regularity at infinity of two variable Pick functions

A Pick function of $d$ variables is a holomorphic map from $Π^d$ to $Π$, where $Π$ is the upper halfplane. Some Pick functions of one variable have an asymptotic expansion at infinity, a power series $\sum_{n=1}^\infty ρ_n z^{-n}$ with real numbers $ρ_n$ that gives an asymptotic expansion on non-tangential approach regions to infinity. H. Hamburger in 1921 characterized which sequences $\{ρ_n\} $ can occur. We give an extension of Hamburger's results to Pick functions of two variables.

math.CV↗
arXiv2011

Pseudo-Taylor expansions and the Carathéodory-Fejér problem

We give a new solvability criterion for the boundary Carathéodory-Fejér problem: given a point $x \in \mathbb{R}$ and, a finite set of target values $a^0,a^1,...,a^n \in \mathbb{R}$, to construct a function $f$ in the Pick class such that the limit of $f^{(k)}(z)/k!$ as $z \to x$ nontangentially in the upper half plane is $a^k$ for $k= 0,1,...,n$. The criterion is in terms of positivity of an associated Hankel matrix. The proof is based on a reduction method due to Julia and Nevanlinna.

math.CV↗
arXiv2010

The boundary Carathéodory-Fejér interpolation problem

We give an elementary proof of a solvability criterion for the {\em boundary Carathéodory-Fejér 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.

math.CV↗
arXiv2010

Facial behaviour of analytic functions on the bidisk

We prove that if $ϕ$ is an analytic function bounded by 1 on the bidisk and $τ$ is a point in a face of the bidisk at which $ϕ$ satisfies Caratheodory's condition then both $ϕ$ and the angular gradient $\nablaϕ$ exist and are constant on the face. Moreover, the class of all $ϕ$ with prescribed $ϕ(τ)$ and $\nablaϕ(τ)$ 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.

math.CV↗
arXiv2010

Boundary Nevanlinna-Pick interpolation via reduction and augmentation

We give an elementary proof of Sarason's solvability criterion for the Nevanlinna-Pick problem with boundary interpolation nodes and boundary target values. We also give a concrete parametrization of all solutions of such a problem. The proofs are based on a reduction method due to Julia and Nevanlinna. Reduction of functions corresponds to Schur complementation of the corresponding Pick matrices.

math.CV↗
arXiv2009

Algebraic pairs of isometries

We consider pairs of commuting isometries that are annihilated by a polynomial. We show that the polynomial must be inner toral, which is a geometric condition on its zero set. We show that cyclic pairs of commuting isometries are nearly unitarily equivalent if they are annihilated by the same minimal polynomial.

math.FA↗
arXiv2007

Cusp Algebras

We consider simple cusp algebras, that is certain subalgebras of the algebra of holomorphic functions on a disk that are annihilated by some distributions living on a singleton. We determine when these algebras can be holized in two dimensions, and when these holizations are globally biholomorphic.

math.AG↗
arXiv2006

Hyperbolic algebraic and analytic curves

A hyperbolic algebraic curve is a bounded subset of an algebraic set. We study the function theory and functional analytic aspects of these sets. We show that their function theory can be described by finite codimensional subalgebras of the holomorphic functions on the desingularization. We show that classical analytic techniques, such as interpolation, can be used to answer geometric questions about the existence of biholomorphic maps. Conversely, we show that the algebraic-geometric viewpoint leads to interesting questions in classical analysis.

math.FA↗