arXiv subjects

Nicholas J. Young

Publications and source records attributed to Nicholas J. Young.

6 recordsLinked to original sources

arXiv2026

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

math.CV↗

arXiv2019

Calcular Algebras

A calcular algebra is a subalgebra of $H^\infty(Ω)$ with norm given by $\| ϕ\| = \sup \| ϕ(T) \|$ as $T$ ranges over a given class of commutative $d$-tuples of operators with Taylor spectrum in $Ø$. We discuss what algebras arise this way, and how they can be represented.

math.FA↗

arXiv2017

Algebraic and geometric aspects of rational $Γ$-inner functions

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 $Γ$.

math.CV↗

arXiv2013

Operator monotone functions and Löwner functions of several variables

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.

math.FA↗

arXiv2013

On the Representation of Holomorphic Functions on Polyhedra

We generalize the Oka extension theorem, and obtain bounds on the norm of the extension, by using operator theory.

math.CV↗

arXiv1995

Continuity properties of best analytic approximation

Let $\A$ be the operator which assigns to each $m \times n$ matrix-valued function on the unit circle with entries in $H^\infty + C$ its unique superoptimal approximant in the space of bounded analytic $m \times n$ matrix-valued functions in the open unit disc. We study the continuity of $\A$ with respect to various norms. Our main result is that, for a class of norms satifying certain natural axioms, $\A$ is continuous at any function whose superoptimal singular values are non-zero and is such that certain associated integer indices are equal to 1. We also obtain necessary conditions for continuity of $\A$ at point and a sufficient condition for the continuity of superoptimal singular values.

math.FA↗