arXiv subjects

N. J. Young

Publications and source records attributed to N. J. Young.

At least 19 recordsLinked to original sources

arXiv2026

Boundary behavior of functions in the Schur-Agler class of the polydisc

We describe a generalization of the notion of a Hilbert space model of a function $φ$ in the Schur-Agler class of the polydisc. This generalization is well adapted to the investigation of boundary behavior of $φ$ at a mild singularity $τ$ on the $d$-torus. We prove the existence of a generalized model with an enhanced continuity property at such a singularity $τ$. We use this result to prove the directional differentiability of a function $φ$ in the Schur-Agler class at a singular point on the $d$-torus for which the Carathéodory condition holds and to calculate the corresponding directional derivative. The results of this paper extend to the polydisc $\mathbb{D}^d$ results of Agler, McCarthy, Tully-Doyle and Young which generalized to the bidisc the classical Julia-Wolff-Carathéodory theorem about analytic self-maps of $\mathbb{D}$.

math.CV↗

arXiv2026

Models of holomorphic functions on the symmetrized skew bidisc

The purpose of this paper is to develop the theory of holomorphic functions with modulus bounded by $1$ on the symmetrized skew bidisc \[ \mathbb{G}_{r} \stackrel{\rm def}{=} \Big\{( λ_{1}+rλ_{2} ,rλ_{1}λ_{2}): λ_{1}\in \mathbb{D}, λ_{2}\in\mathbb{D}\Big\}, \] for a fixed $r \in (0,1)$. We show the existence of a realization formula and a model formula for such holomorphic functions.

math.CV↗

arXiv2025

Function theory on the annulus in the dp-norm

In this paper we shall use realization theory to prove new results about a class of holomorphic functions on an annulus \[R_δ\stackrel{\rm def}{=} \{z \in \mathbb{C}: δ<|z|<1\},\] where $0<δ<1$. The class of functions in question arises in the early work of R. G. Douglas and V. I. Paulsen on the rational dilation of a Hilbert space operator $T$ to a normal operator with spectrum in $\partial R_δ$. Their work suggested the following norm $\|\cdot\|_{\mathrm{dp}}$ on the space $\mathrm{Hol}(R_δ)$ of holomorphic functions on $R_δ$, \[ \|ϕ\|_{\mathrm{dp}} \stackrel{\rm def}{=} \sup\{ \|ϕ(T)\|: \|T\|\leq 1, \|T^{-1} \|\leq 1/δ\ \text{and} \ σ(T)\subseteq R_δ\}.\] By analogy with the classical Schur class of holomorphic functions $\mathcal{S} $ with supremum norm at most $1$ on the disc $\mathbb{D}$, it is natural to consider the dp-Schur class $\mathcal{S}_\mathrm{dp}$ of holomorphic functions of dp-norm at most $1$ on $R_δ$. Our central result is a Pick interpolation theorem for functions in $\mathcal{S}_\mathrm{dp}$ that is analogous to Abrahamse's Interpolation Theorem for bounded holomorphic functions on a multiply-connected domain. For a tuple $λ=(λ_1,\dots,λ_n)$ of distinct interpolation nodes in $R_δ$, we introduce a special set $\mathcal{G}_{\mathrm {dp}}(λ)$ of positive definite $n\times n$ matrices, which we call DP Szegő kernels. The DP Pick problem $λ_j \mapsto z_j, j=1,\dots,n$, is shown to be solvable if and only if, \[ [(1-\bar z_i z_j)g_{ij}] \ge 0 \; \text{ for all}\; g \in \mathcal{G}_{\mathrm {dp}} (λ).\] We prove further that a solvable DP Pick problem has a solution which is a rational function.

math.CV↗

arXiv2024

On the Operators with Numerical Range in an Ellipse

We give new necessary and sufficient conditions for the numerical range $W(T)$ of an operator $T \in \mathcal{B}(\mathcal{H})$ to be a subset of the closed elliptical set $K_δ\subseteq \mathbb{C}$ given by \[ K_δ{\stackrel{\rm def}{=}} \left\{x+iy: \frac{x^2}{(1+δ)^2} + \frac{y^2}{(1-δ)^2} \leq 1\right\}, \] where $0 < δ< 1$. Here $\mathcal{B}(\mathcal{H})$ denotes the collection of bounded linear operators on a Hilbert space $\mathcal{H}$. Central to our efforts is a direct generalization of Berger's well-known criterion for an operator to have numerical radius at most one, his so-called strange dilation theorem. We next generalize the lemma of Sarason that describes power dilations in terms of semi-invariant subspaces to operators $T$ that satisfy appropriate dilation properties. This generalization yields a characterization of the operators $T\in \mathcal{B}(\mathcal{H})$ such that $W(T)$ is contained in $K_δ$ in terms of certain structured contractions that act on $\mathcal{H} \oplus \mathcal{H}$. As a corollary of our results we extend Ando's parametrization of operators having numerical range in a disc to those $T$ such that $W(T)\subseteq K_δ$. We prove that, if $T$ acts on a finite-dimensional Hilbert space $\mathcal{H}$, then $W(T)\subseteq K_δ$ if and only if there exist a pair of contractions $A,B \in \mathcal{B}(\mathcal{H})$ such that $A$ is self-adjoint and \[ T=2\sqrtδA + (1-δ)\sqrt{1+A}\ B\sqrt{1-A}. \] We also obtain a formula for the B. and F. Delyon calcular norm of an analytic function on the inside of an ellipse in terms of the extremal $H^\infty$-extension problem for analytic functions defined on a slice of the symmetrized bidisc.

math.FA↗

arXiv2024

Non-commutative manifolds, the free square root and symmetric functions in two non-commuting variables

The richly developed theory of complex manifolds plays important roles in our understanding of holomorphic functions in several complex variables. It is natural to consider manifolds that will play similar roles in the theory of holomorphic functions in several non-commuting variables. In this paper we introduce the class of \emph{nc-manifolds}, the mathematical objects that at each point possess a neighborhood that has the structure of an \emph{nc-domain} in the \emph{$d$-dimensional nc-universe $\m^d$}. We illustrate the use of such manifolds in free analysis through the construction of the non-commutative Riemann surface for the matricial square root function. A second illustration is the construction of a non-commutative analog of the elementary symmetric functions in two variables. For any symmetric domain in $\m^2$ we construct a 2-dimensional non-commutative manifold such that the symmetric holomorphic functions on the domain are in bijective correspondence with the holomorphic functions on the manifold. We also derive a version of the classical Newton-Girard formulae for power sums of two non-commuting variables.

math.CV↗

arXiv2022

A Hilbert space approach to singularities of functions

We introduce the notion of a pseudomultiplier of a Hilbert space $\mathcal H$ of functions on a set $Ω$. Roughly, a pseudomultiplier of $\mathcal H$ is a function which multiplies a finite-codimensional subspace of $\mathcal H$ into $\mathcal H$, where we allow the possibility that a pseudomultiplier is not defined on all of $Ω$. A pseudomultiplier of $\mathcal H$ has singularities, which comprise a subspace of $\mathcal H$, and generalize the concept of singularities of an analytic function, even though the elements of $\mathcal H$ need not enjoy any sort of analyticity. We analyse the natures of these singularities, and obtain a broad classification of them in function-theoretic terms.

math.FA↗

arXiv2022

Nonuniqueness of Carathéodory extremal functions on the symmetrized bidisc

We survey the Carathéodory extremal problem $\mathrm{Car} δ$ on the symmetrized bidisc $$ G = \{(z+w,zw):|z|<1, \, |w|<1\} = \{(s,p)\in \mathbb{C}^2: |s-\bar s p| < 1-|p|^2\}. $$ We also give some new results on this topic. We are particularly interested in cases of this problem in which the solution of the problem is not unique. It is known that, for any $δ=(λ,v)\in TG$ with $v\neq 0$, there is at least one $ω\in\mathbb{T}$ such that $Φ_ω$ solves $\mathrm{Car} δ$, where $Φ_ω(s,p) = \frac{2ωp-s}{2-ωs}$. Moreover, there is an essentially unique solution of $\mathrm{Car} δ$ if and only if $δ$ has exactly one Carathéodory extremal function of the form $Φ_ω$ for some $ω\in\mathbb{T}$. We give a description of Carathéodory extremals for $δ\in TG$ with more than one Carathéodory extremal function $Φ_ω$ for some values of $ω\in\mathbb{T}$. The proof exploits a model formula for the Schur class of $G$ which is an analog of the well-known network realization formula for Schur-class functions on the disc.

math.CV↗

arXiv2020

Intrinsic Directions, Orthogonality and Distinguished Geodesics in the Symmetrized Bidisc

The symmetrized bidisc \[ G \stackrel{\rm{def}}{=}\{(z+w,zw):|z|<1,\ |w|<1\}, \] under the Carathéodory metric, is a complex Finsler space of cohomogeneity $1$ in which the geodesics, both real and complex, enjoy a rich geometry. As a Finsler manifold, $G$ does not admit a natural notion of angle, but we nevertheless show that there {\em is} a notion of orthogonality. The complex tangent bundle $TG$ splits naturally into the direct sum of two line bundles, which we call the {\em sharp} and {\em flat} bundles, and which are geometrically defined and therefore covariant under automorphisms of $G$. Through every point of $G$ there is a unique complex geodesic of $G$ in the flat direction, having the form \[ F^β\stackrel{\rm{def}}{=}\{(β+\barβz,z)\ : z\in\mathbb{D}\} \] for some $β\in\mathbb{D}$, and called a {\em flat geodesic}. We say that a complex geodesic \emph{$D$ is orthogonal} to a flat geodesic $F$ if $D$ meets $F$ at a point $λ$ and the complex tangent space $T_λD$ at $λ$ is in the sharp direction at $λ$. We prove that a geodesic $D$ has the closest point property with respect to a flat geodesic $F$ if and only if $D$ is orthogonal to $F$ in the above sense. Moreover, $G$ is foliated by the geodesics in $G$ that are orthogonal to a fixed flat geodesic $F$.

math.DG↗

arXiv2020

Exterior powers and pointwise creation operators

We develop a theory of pointwise wedge products of vector-valued functions on the circle and the disc, and obtain results which give rise to a new approach to the analysis of the matricial Nehari problem. We investigate properties of pointwise creation operators and pointwise orthogonal complements in the context of operator theory and the study of vector-valued analytic functions on the unit disc.

math.CV↗

arXiv2019

A Geometric Characterization of the Symmetrized Bidisc

The symmetrized bidisc \[ G \stackrel{\rm{def}}{=}\{(z+w,zw):|z|<1,\ |w|<1\} \] has interesting geometric properties. While it has a plentiful supply of complex geodesics and of automorphisms, there is nevertheless a unique complex geodesic $\mathcal{R}$ in $G$ that is invariant under all automorphisms of $G$. Moreover, $G$ is foliated by those complex geodesics that meet $\mathcal{R}$ in one point and have nontrivial stabilizer. We prove that these properties, together with two further geometric hypotheses on the action of the automorphism group of $G$, characterize the symmetrized bidisc in the class of complex manifolds.

math.CV↗

arXiv2018

Characterizations of some domains via Carathéodory extremals

In this paper we characterize the unit disc, the bidisc and the symmetrized bidisc \[ G =\{(z+w,zw):|z|<1,\ |w|<1\} \] in terms of the possession of small classes of analytic maps into the unit disc that suffice to solve all Carathéodory extremal problems in the domain.

math.CV↗

arXiv2018

Carathéodory extremal functions on the symmetrized bidisc

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.

math.CV↗

arXiv2018

Analytic interpolation into the tetrablock and a $μ$-synthesis problem

We give a solvability criterion for a special case of the $μ$-synthesis problem. That is, we prove the necessity and sufficiency of a condition for the existence of an analytic $2 \times 2$ matrix-valued function on the disc subject to a bound on the structured singular value and satisfying a finite set of interpolation conditions. To do this we prove a realization theorem for analytic functions from the disc to the tetrablock. We also obtain a solvability criterion for the problem of analytic interpolation from the disc to the tetrablock.

math.CV↗

arXiv2017

Realization of functions on the symmetrized bidisc

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.

math.CV↗

arXiv2016

A rich structure related to the construction of analytic matrix functions

We analyse two special cases of $μ$-synthesis problems which can be reduced to interpolation problems in the set of analytic functions from the disc into the symmetrised bidisc and into the tetrablock. For these inhomogeneous domains we study the structure of interconnections between the set of analytic functions from the disc into the given domain, the matricial Schur class, the Schur class of the bidisc, and the set of pairs of positive kernels on the bidisc subject to a boundedness condition. We use the theories of Hilbert function spaces and of reproducing kernels to establish these connections. We give a solvability criterion for the interpolation problem that arises from the $μ$-synthesis problem related to the tetrablock.

math.CV↗

arXiv2016

Finite Blaschke products and the construction of rational $Γ$-inner functions

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.

math.CV↗

arXiv2015

LOFAR discovery of a quiet emission mode in PSR B0823+26

PSR B0823+26, a 0.53-s radio pulsar, displays a host of emission phenomena over timescales of seconds to (at least) hours, including nulling, subpulse drifting, and mode-changing. Studying pulsars like PSR B0823+26 provides further insight into the relationship between these various emission phenomena and what they might teach us about pulsar magnetospheres. Here we report on the LOFAR discovery that PSR B0823+26 has a weak and sporadically emitting 'quiet' (Q) emission mode that is over 100 times weaker (on average) and has a nulling fraction forty-times greater than that of the more regularly-emitting 'bright' (B) mode. Previously, the pulsar has been undetected in the Q-mode, and was assumed to be nulling continuously. PSR B0823+26 shows a further decrease in average flux just before the transition into the B-mode, and perhaps truly turns off completely at these times. Furthermore, simultaneous observations taken with the LOFAR, Westerbork, Lovell, and Effelsberg telescopes between 110 MHz and 2.7 GHz demonstrate that the transition between the Q-mode and B-mode occurs within one single rotation of the neutron star, and that it is concurrent across the range of frequencies observed.

astro-ph.SR↗

arXiv2015

Long-term Observations of Three Nulling Pulsars

We present an analysis of approximately 200 hours of observations of the pulsars J1634$-$5107, J1717$-$4054 and J1853$+$0505, taken over the course of 14.7 yr. We show that all of these objects exhibit long term nulls and radio-emitting phases (i.e. minutes to many hours), as well as considerable nulling fractions (NFs) in the range $\sim67\,\% - 90\,\%$. PSR J1717$-$4054 is also found to exhibit short timescale nulls ($1 - 40~P$) and burst phases ($\lesssim 200~P$) during its radio-emitting phases. This behaviour acts to modulate the NF, and therefore the detection rate of the source, over timescales of minutes. Furthermore, PSR J1853$+$0505 is shown to exhibit a weak emission state, in addition to its strong and null states, after sufficient pulse integration. This further indicates that nulls may often only represent transitions to weaker emission states which are below the sensitivity thresholds of particular observing systems. In addition, we detected a peak-to-peak variation of $33\pm1\,\%$ in the spin-down rate of PSR J1717$-$4054, over timescales of hundreds of days. However, no long-term correlation with emission variation was found.

astro-ph.HE↗