arXiv subjects
Publications and source records attributed to Jim Agler.
We show how realization theory can be used to find the solutions of the Carathéodory 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 Carathéodory problems for tangents of non-generic types.
We show that if the minimum entropy for a polynomial with roots on the unit circle is attained by polynomials with equally spaced roots, then, under a generic hypothesis about the nature of the extremum, the Krzyz conjecture on the maximum modulus of the Taylor coefficients of a holomorphic function that maps the disk to the punctured disk is true.
The set \[ Γ{\stackrel{\rm def}{=}} \{(z+w,zw):|z|\leq 1,|w|\leq 1\} \subset {\mathbb{C}}^2 \] has intriguing complex-geometric properties; it has a 3-parameter group of automorphisms, its distinguished boundary is a ruled surface homeomorphic to the Möbius 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 $Γ$ that map the boundary of the disc to the distinguished boundary of $Γ$.
We discuss non commutative functions, which naturally arise when dealing with functions of more than one matrix variable.
We give a new proof that bounded non-commutative functions on polynomial polyhedra can be represented by a realization formula, a generalization of the transfer function realization formula for bounded analytic functions on the unit disk.
We prove a Montel theorem for Hilbert space valued functions, and a non-commutative version of this theorem, by composing with unitaries to achieve convergence.
We prove a realization formula and a model formula for analytic functions with modulus bounded by $1$ on the symmetrized bidisc \[ G\stackrel{\rm def}{=} \{(z+w,zw): |z|<1, \, |w| < 1\}. \] As an application we prove a Pick-type theorem giving a criterion for the existence of such a function satisfying a finite set of interpolation conditions.
Let \[ Γ= \{(z+w, zw): |z|\leq 1, |w|\leq 1\} \subset \mathbb{C}^2. \] A $Γ$-inner function is defined to be a holomorphic map $h$ from the unit disc $\mathbb{D}$ to $Γ$ whose boundary values at almost all points of the unit circle $\mathbb{T}$ belong to the distinguished boundary $bΓ$ of $Γ$. A rational $Γ$-inner function $h$ induces a continuous map $h|_\mathbb{T}$ from the unit circle to $bΓ$. The latter set is topologically a Möbius band and so has fundamental group $\mathbb{Z}$. The {\em degree} of $h$ is defined to be the topological degree of $h|_\mathbb{T}$. In a previous paper the authors showed that if $h=(s,p)$ is a rational $Γ$-inner function of degree $n$ then $s^2-4p$ has exactly $n$ zeros in the closed unit disc $\mathbb{D}^-$, counted with an appropriate notion of multiplicity. In this paper, with the aid of a solution of an interpolation problem for finite Blaschke products, we explicitly construct the rational $Γ$-inner functions of degree $n$ with the $n$ zeros of $s^2-4p$ and the corresponding values of $s$, prescribed.
We give a characterization of interpolating sequences for bounded analytic functions on the bidisk.
We analyze the three point Pick interpolation problem on the bidisk
We describe those reproducing kernel Hilbert spaces of holomorphic functions on domains in ${\Bbb C}^d$ for which an analogue of the Nevanlinna-Pick theorem holds, in other words when the existence of a (possibly matrix-valued) function in the unit ball of the multiplier algebra with specified values on a finite set of points is equivalent to the positvity of a related matrix. Our description is in terms of a certain localization property of the kernel.
We give a new treatment of Quiggin's and McCullough's characterization of complete Nevanlinna-Pick kernels. We show that a kernel has the matrix-valued Nevanlinna-Pick property if and only if it has the vector-valued Nevanlinna-Pick property. We give a representation of all complete Nevanlinna-Pick kernels, and show that they are all restrictions of a universal complete Nevanlinna-Pick kernel.
A set $V$ in a domain $U$ in $\mathbb{C}^n$ has the {\em norm-preserving extension property} if every bounded holomorphic function on $V$ has a holomorphic extension to $U$ with the same supremum norm. We prove that an algebraic subset of the {\em symmetrized bidisc} \[ G := \{(z+w,zw):|z|<1, |w| < 1 \} \] has the norm-preserving extension property if and only if it is either a singleton, $G$ itself, a complex geodesic of $G$, or the union of the set $\{(2z,z^2): |z|<1\}$ and a complex geodesic of degree $1$ in $G$. We also prove that the complex geodesics in $G$ coincide with the nontrivial holomorphic retracts in $G$. Thus, in contrast to the case of the ball or the bidisc, there are sets in $G$ which have the norm-preserving extension property but are not holomorphic retracts of $G$. In the course of the proof we obtain a detailed classification of the complex geodesics in $G$ modulo automorphisms of $G$. We give applications to von Neumann-type inequalities for $Γ$-contractions (that is, commuting pairs of operators for which the closure of $G$ 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 $2\times 2$ matrices, the tetrablock and the pentablock. We also identify the subsets of the bidisc which have the norm-preserving extension property for symmetric functions.
We characterize functions of $d$-tuples of bounded operators on a Hilbert space that are uniformly approximable by free polynomials on balanced open sets.
We develop a functional calculus for $d$-tuples of non-commuting elements in a Banach algebra. The functions we apply are free analytic functions, that is nc functions that are bounded on certain polynomial polyhedra.
We prove an implicit function theorem for non-commutative functions. We use this to show that if $p(X,Y)$ is a generic non-commuting polynomial in two variables, and $X$ is a generic matrix, then all solutions $Y$ of $p(X,Y)=0$ will commute with $X$.
We prove generalizations of Löwner'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.
We prove an interpolation theorem for bounded free holomorphic functions.