arXiv subjects

Ryan Tully-Doyle

Publications and source records attributed to Ryan Tully-Doyle.

17 recordsLinked to original sources

arXiv2026

Automorphic Nelson Dilations for Contractions and Invariant Subspace Tracking

Given an $n \times n$ strictly contractive matrix $T$, an (automorphic) Nelson dilation $\widehat{T}$ of $T$ is a certain type of analytic matrix-valued function on the unit disk with $\widehat{T}(0) = T$. Its construction gives a method for lifting a matrix to a matrix-valued function with nice boundary behavior, a trick that has proved useful in recent operator theoretic developments. In this paper, we show that Nelson dilations give a quick way to obtain the minimal isometric and unitary dilations of $T$ and thus, connect naturally to the classical Sz.-Nagy dilation theory. We then initiate the study of the automorphic Nelson dilations as a fundamental object in their own right and prove that every $T$ has Nelson dilations $\widehat{T}$ with particularly useful/interesting properties; for example, they either have strongly entangled eigenvalue functions or have reducing subspaces that are independent of $z$. Along the way, we examine when the product of an invertible matrix and a diagonal matrix has distinct eigenvalues.

math.FA↗

arXiv2024

Cauchy Transforms of Colored Graphs in Two Variables

By designating vertices with variables, a simple undirected graph can be augmented to have an associated representing rational function in two variables taking the complex bi-upper halfplane to itself. We give relations between representing functions of certain products of such graphs by way of Schur complements. We also study the connection between the structure of the graph and the regularity of the representing function at a boundary singularity.

math.CV↗

arXiv2024

Matrix convex verbatim enumeration functions are graphical

We give a relation between verbatim generating functions of what we call Pythagorean languages and matrix convexity. Namely, several multivariate matrix convex functions occurring in the existing matrix analysis literature arise naturally in a combinatorial way. We give a Gelfand type formula for the numerical radius.

math.CO↗

arXiv2022

Induced Stinespring factorization and the Wittstock support theorem

Given a pair of self-adjoint-preserving completely bounded maps on the same $C^*$-algebra, say that $φ\leq ψ$ if the kernel of $φ$ is a subset of the kernel of $ψ$ and $ψ\circ φ^{-1}$ is completely positive. The \emph{Agler class} of a map $φ$ is the class of $ψ\geq φ.$ Such maps admit colligation formulae, and, in Lyapunov type situations, transfer function type realizations on the Stinespring coefficients of their Wittstock decompositions. As an application, we prove that the support of an extremal Wittstock decomposition is unique.

math.OA↗

arXiv2021

Dynamics of low-degree rational inner skew-products on $\mathbb{T}^2$

We examine iteration of certain skew-products on the bidisk whose components are rational inner functions, with emphasis on simple maps of the form $\Phi(z_1,z_2) = (\phi(z_1,z_2), z_2)$. If $\phi$ has degree $1$ in the first variable, the dynamics on each horizontal fiber can be described in terms of M\"obius transformations but the global dynamics on the $2$-torus exhibit some complexity, encoded in terms of certain $\mathbb{T}^2$-symmetric polynomials. We describe the dynamical behavior of such mappings $\Phi$ and give criteria for different configurations of fixed point curves and rotation belts in terms of zeros of a related one-variable polynomial.

math.DS↗

arXiv2021

Averaged mixed Julia-Fatou type theory with applications to spectral foliation

Classically, theorems of Fatou and Julia describe the boundary regularity of functions in one complex variable. The former says that a complex analytic function on the disk has non-tangential boundary values almost everywhere, and the latter describes when a function takes an extreme value at a boundary point and is differentiable there non-tangentially. We describe a class of intermediate theorems in terms of averaged Julia-Fatou quotients. Boundary regularity is related to integrability of certain quantities against a special measure, the so-called Nevanlinna measure. Applications are given to spectral theory.

math.CV↗

arXiv2021

Monotonicity of the principal pivot transform

We prove that the principal pivot transform (also known as the partial inverse, sweep operator, or exchange operator in various contexts) maps matrices with positive imaginary part to matrices with positive imaginary part. We show that the principal pivot transform is matrix monotone by establishing Hermitian square representations for the imaginary part and the derivative.

math.FA↗

arXiv2020

Analytic continuation of concrete realizations and the McCarthy Champagne conjecture

In this paper, we give formulas that allow one to move between transfer function type realizations of multi-variate Schur, Herglotz and Pick functions, without adding additional singularities except perhaps poles coming from the conformal transformation itself. In the two-variable commutative case, we use a canonical de Branges-Rovnyak model theory to obtain concrete realizations that analytically continue through the boundary for inner functions which are rational in one of the variables (so-called quasi-rational functions). We then establish a positive solution to McCarthy's Champagne conjecture for local to global matrix monotonicity in the settings of both two-variable quasi-rational functions and $d$-variable perspective functions.

math.FA↗

arXiv2019

Automatic real analyticity and a regal proof of a commutative multivariate Löwner theorem

We adapt the "royal road" method used to simplify automatic analyticity theorems in noncommutative function theory to several complex variables. We show that certain families of functions must be real analytic if they have certain nice properties on one dimensional slices. Let $E \subset \mathbb{R}^d$ be open. A function $f:E \to \mathbb{R}$ is matrix monotone lite if $f(φ_1(t), \ldots, φ_d(t))$ is a matrix monotone function of $t$ whenever $t \in (0,1)$, the $φ_i$ are automorphisms of the upper half plane, and the tuple $(φ_1(t), \ldots, φ_d(t))$ maps $(0,1)$ into $E$. We use the "royal road" to show that a function is matrix monotone lite if and only if it analytically continues to the multi-variate upper half plane as a map into the upper half plane. Moreover, matrix monotone lite functions in two variables are locally matrix monotone in the sense of Agler-McCarthy-Young.

math.FA↗

arXiv2019

The royal road to automatic noncommutative real analyticity, monotonicity, and convexity

It was shown classically that matrix monotone and matrix convex functions must be real analytic by Löwner and Kraus respectively. Recently, various analogues have been found in several noncommuting variables. We develop a general framework for lifting automatic analyticity theorems in matrix analysis from one variable to several variables, the so-called "royal road theorem." That is, we establish the principle that the hard part of proving any automatic analyticity theorem lies in proving the one variable theorem. We use our main result to prove the noncommutative Löwner and Kraus theorems over operator systems as examples, including an analogue of the "butterfly realization" of Helton-McCullough-Vinnikov for general analytic functions.

math.FA↗

arXiv2018

Escaping nontangentiality: Towards a controlled tangential amortized Julia-Carathéodory theory

Let $f: D \rightarrow Ω$ be a complex analytic function. The Julia quotient is given by the ratio between the distance of $f(z)$ to the boundary of $Ω$ and the distance of $z$ to the boundary of $D.$ A classical Julia-Carathéodory type theorem states that if there is a sequence tending to $τ$ in the boundary of $D$ along which the Julia quotient is bounded, then the function $f$ can be extended to $τ$ such that $f$ is nontangentially continuous and differentiable at $τ$ and $f(τ)$ is in the boundary of $Ω.$ We develop an extended theory when $D$ and $Ω$ are taken to be the upper half plane which corresponds to amortized boundedness of the Julia quotient on sets of controlled tangential approach, so-called $λ$-Stolz regions, and higher order regularity, including but not limited to higher order differentiability, which we measure using $γ$-regularity. Applications are given, including perturbation theory and moment problems.

math.FA↗

arXiv2018

Representation of free Herglotz functions

A Herglotz function is a holomorphic map from the open complex unit disk into the closed complex right halfplane. A classical Herglotz function has an integral representation against a positive measure on the unit circle. We prove a free analytic analogue of the Herglotz representation and describe how our representations specialize to the free probabilistic case. We also show that the set of representable Herglotz functions arising from noncommutative conditional expectations must be closed in a natural topology.

math.OA↗

arXiv2016

Cauchy transforms arising from homomorphic conditional expectations parametrize free Pick functions but those arising from conditional expectations do not

Nevanlinna showed that Cauchy transforms of probability measures parametrize all functions from the upper half plane into itself satisfying a certain asymptotic condition at infinity. We show that the correspondence fails in general for the unbounded case for somewhat trivial reasons; however, we show that in a setting of "homomorphic" operator valued free probability that Cauchy transforms of homomorphic conditional expectations parametrize free Pick functions.

math.FA↗

arXiv2016

Analytic functions on the bidisk at boundary singularities via Hilbert space methods

We investigate the behavior of a generalized Hilbert space model of a function in the Schur class of the bidisk at singular boundary points that satisfy a growth condition. We examine the relationship between the boundary behavior of Schur functions and the geometry of corresponding generalized Hilbert space models. We describe a geometric condition on an associated operator that classifies the behavior of the directional derivative of the underlying Schur function at a carapoint.

math.FA↗

arXiv2015

Convex entire noncommutative functions are polynomials of degree two or less

This paper concerns matrix "convex" functions of (free) noncommuting variables, $x = (x_1, \ldots, x_g)$. Helton and McCullough showed that a polynomial in $x$ which is matrix convex is of degree two or less. We prove a more general result: that a function of $x$ that is matrix convex near $0$ and also that is "analytic" in some neighborhood of the set of all self-adjoint matrix tuples is in fact a polynomial of degree two or less. More generally, we prove that a function $F$ in two classes of noncommuting variables, $a = (a_1, \ldots, a_{\tilde{g}})$ and $x = (x_1, \ldots, x_g)$ that is "analytic" and matrix convex in $x$ on a "noncommutative open set" in $a$ is a polynomial of degree two or less.

math.FA↗

arXiv2014

Free functions with symmetry

In 1936, Margarete C. Wolf showed that the ring of symmetric free polynomials in two or more variables is isomorphic to the ring of free polynomials in infinitely many variables. We show that Wolf's theorem is a special case of a general theory of the ring of invariant free polynomials: every ring of invariant free polynomials is isomorphic to a free polynomial ring. Furthermore, we show that this isomorphism extends to the free functional calculus as a norm-preserving isomorphism of function spaces on a domain known as the row ball. We give explicit constructions of the ring of invariant free polynomials in terms of representation theory and develop a rudimentary theory of their structures. Specifically, we obtain a generating function for the number of basis elements of a given degree and explicit formulas for good bases in the abelian case.

math.FA↗

arXiv2013

Free Pick functions: representations, asymptotic behavior and matrix monotonicity in several noncommuting variables

We extend the study of the Pick class, the set of complex analytic functions taking the upper half plane into itself, to the noncommutative setting. R. Nevanlinna showed that elements of the Pick class have certain integral representations which reflect their asymptotic behavior at infinity. Loewner connected the Pick class to matrix monotone functions. We generalize the Nevanlinna representation theorems and Loewner's theorem on matrix monotone functions to the free Pick class, the collection of functions that map tuples of matrices with positive imaginary part into the matrices with positive imaginary part which obey the free functional calculus.

math.FA↗