arXiv subjects

Kelly Bickel

Publications and source records attributed to Kelly Bickel.

At least 19 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↗

arXiv2026

Two-Variable Compressions of Shifts, Toeplitz Operators, and Numerical Ranges

This paper studies two-variable compressions of shifts associated to rational inner functions on the bidisk; these generalize the classical compressions of the shift associated to finite Blasckhe products and are unitarily equivalent to one-variable, matrix-valued Toeplitz operators. This paper proves that a rational inner function is almost completely determined by these Toeplitz operator symbols but provides examples showing that (unlike in the one-variable case) rational inner functions are not determined by the numerical ranges of their compressed shifts. This paper also investigates related questions including methods of constructing these compressed-shift Toeplitz operators and when the associated numerical ranges are open and closed.

math.CV↗

arXiv2025

Pairs of Clark Unitary Operators on the Bidisk and their Taylor Joint Spectra

We develop a Clark theory for commuting compressed shift operators on model spaces $K_{\phi}$ associated with inner functions $\phi$ on the bidisk, which exhibits both similarities and marked differences compared to the classical one-variable version. We first identify the adjoint of the embedding operator $J_{\alpha} \colon K_{\phi}\to L^2(\sigma_{\alpha})$ as a weighted Cauchy transform of the Clark measure $\sigma_{\alpha}$. Under natural assumptions, which generically include the case when $\phi$ is rational inner, we obtain commuting unitaries on $K_{\phi}$ that are (often infinite-dimensional) perturbations of the compressed shift operators $K_{\phi}$. We prove that these unitaries are unitarily equivalent to multiplication by the coordinate functions on $L^2(\sigma_\alpha)$ and then establish a number of related properties and simplified results in special cases. Finally, we show that the Taylor joint spectrum of these Clark unitaries coincides with level sets of $\phi$ when $\phi$ is a rational inner function.

math.CV↗

arXiv2025

Stable Polynomials via Undirected Colored Graphs

This paper initiates a systematic study of connections between undirected colored graphs and associated two-variable stable polynomials obtained via Cauchy transform-type formulas. Examples of such stable polynomials have played crucial roles in other recent studies, though their general properties have remained rather opaque. Using linear algebra techniques, this paper characterizes when these stable polynomials have boundary zeros and studies the finer behavior, called contact order, of their zero sets near the guaranteed boundary zeros.

math.CV↗

arXiv2024

Stable polynomials and admissible numerators in product domains

Given a polynomial $p$ with no zeros in the polydisk, or equivalently the poly-upper half-plane, we study the problem of determining the ideal of polynomials $q$ with the property that the rational function $q/p$ is bounded near a boundary zero of $p$. We give a complete description of this ideal of numerators in the case where the zero set of $p$ is smooth and satisfies a non-degeneracy condition. We also give a description of the ideal in terms of an integral closure when $p$ has an isolated zero on the distinguished boundary. Constructions of multivariate stable polynomials are presented to illustrate sharpness of our results and necessity of our assumptions.

math.CV↗

arXiv2023

Crouzeix's conjecture, compressions of shifts, and classes of nilpotent matrices

This paper studies the level set Crouzeix conjecture, which is a weak version of Crouzeix's conjecture that applies to finite compressions of the shift. Amongst other results, this paper establishes the level set Crouzeix conjecture for several classes of $3\times3$, $4\times4$, and $5\times5$ matrices associated to compressions of the shift via a geometric analysis of their numerical ranges. This paper also establishes Crouzeix's conjecture for several classes of nilpotent matrices whose studies are motivated by related compressions of shifts.

math.FA↗

arXiv2023

Clark measures for rational inner functions II: general bidegrees and higher dimensions

We study Clark measures associated with general two-variable rational inner functions (RIFs) on the bidisk, including those with singularities, and with general $d$-variable rational inner functions with no singularities. We give precise descriptions of support sets and weights for such Clark measures in terms of level sets and partial derivatives of the associated RIF. In two variables, we characterize when the associated Clark embeddings are unitary, and for generic parameter values, we relate vanishing of two-variable weights with the contact order of the associated RIF at a singularity.

math.FA↗

arXiv2021

Blaschke Products, Level Sets, and Crouzeix's Conjecture

We study several problems motivated by Crouzeix's conjecture, which we consider in the special setting of model spaces and compressions of the shift with finite Blaschke products as symbols. We pose a version of the conjecture in this setting, called the level set Crouzeix (LSC) conjecture, and establish structural and uniqueness properties for (open) level sets of finite Blaschke products that allow us to prove the LSC conjecture in several cases. In particular, we use the geometry of the numerical range to prove the LSC conjecture for compressions of the shift corresponding to unicritical Blaschke products of degree $4$.

math.FA↗

arXiv2021

Local theory of stable polynomials and bounded rational functions of several variables

We provide detailed local descriptions of stable polynomials in terms of their homogeneous decompositions, Puiseux expansions, and transfer function realizations. We use this theory to first prove that bounded rational functions on the polydisk possess non-tangential limits at every boundary point. We relate higher non-tangential regularity and distinguished boundary behavior of bounded rational functions to geometric properties of the zero sets of stable polynomials via our local descriptions. For a fixed stable polynomial $p$, we analyze the ideal of numerators $q$ such that $q/p$ is bounded on the bi-upper half plane. We completely characterize this ideal in several geometrically interesting situations including smooth points, double points, and ordinary multiple points of $p$. Finally, we analyze integrability properties of bounded rational functions and their derivatives on the bidisk.

math.CV↗

arXiv2021

Zero-free regions near a line

We analyze metrics for how close an entire function of genus one is to being real rooted. These metrics arise from truncated Hankel matrix positivity-type conditions built from power series coefficients at each real point. Specifically, if such a function satisfies our positivity conditions and has well-spaced zeros, we show that all of its zeros have to (in some explicitly quantified sense) be far away from the real axis. The obvious interesting example arises from the Riemann zeta function, where our positivity conditions yield a family of relaxations of the Riemann hypothesis. One might guess that as we tighten our relaxation, the zeros of the zeta function must be close to the critical line. We show that the opposite occurs: any potential complex zeros are forced to be farther and farther away from the critical line.

math.CV↗

arXiv2021

Clark Measures for Rational Inner Functions

We analyze the fine structure of Clark measures and Clark isometries associated with two-variable rational inner functions on the bidisk. In the degree (n,1) case, we give a complete description of supports and weights for both generic and exceptional Clark measures, characterize when the associated embedding operators are unitary, and give a formula for those embedding operators. We also highlight connections between our results and both the structure of Agler decompositions and study of extreme points for the set of positive pluriharmonic measures on 2-torus.

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↗

arXiv2020

Crouzeix's Conjecture and related problems

In this paper, we establish several results related to Crouzeix's conjecture. We show that the conjecture holds for contractions with eigenvalues that are sufficiently well-separated. This separation is measured by the so-called separation constant, which is defined in terms of the pseudohyperbolic metric. Moreover, we study general properties of related extremal functions and associated vectors. Throughout, compressions of the shift serve as illustrating examples which also allow for refined results.

math.FA↗

arXiv2019

Singularities of rational inner functions in higher dimensions

We study the boundary behavior of rational inner functions (RIFs) in dimensions three and higher from both analytic and geometric viewpoints. On the analytic side, we use the critical integrability of the derivative of a rational inner function of several variables to quantify the behavior of a RIF near its singularities, and on the geometric side we show that the unimodular level sets of a RIF convey information about its set of singularities. We then specialize to three-variable degree $(m,n,1)$ RIFs and conduct a detailed study of their derivative integrability, zero set and unimodular level set behavior, and non-tangential boundary values. Our results, coupled with constructions of non-trivial RIF examples, demonstrate that much of the nice behavior seen in the two-variable case is lost in higher dimensions.

math.CV↗

arXiv2019

Two-Weight Tb Theorems for Well-Localized Operators

This paper first defines operators that are "well-localized" with respect to a pair of accretive functions and establishes a global two-weight Tb theorem for such operators. Then it defines operators that are "well-localized" with respect to a pair of accretive systems and establishes a local two-weight Tb theorem for them. The proofs combine recent Tb proof techniques with arguments used to prove earlier T1 theorems for well-localized operators.

math.CA↗

arXiv2018

Applications of Envelopes

Intuitively, an envelope of a family of curves is a curve that is tangent to a member of the family at each point. Here we use envelopes of families of circles to study objects from matrix theory and hyperbolic geometry. First we explore relationships between numerical ranges of $2\times 2$ matrices and families of circles to study the elliptical range theorem. Then we deduce a relationship between envelopes and the boundaries of families of intersecting circles and use it to find the boundaries of various families of pseudohyperbolic disks.

math.FA↗

arXiv2018

Numerical Range and Compressions of the Shift

The numerical range of a bounded, linear operator on a Hilbert space is a set in $\mathbb{C}$ that encodes important information about the operator. In this survey paper, we first consider numerical ranges of matrices and discuss several connections with envelopes of families of curves. We then turn to the shift operator, perhaps the most important operator on the Hardy space $H^2(\mathbb{D})$, and compressions of the shift operator to model spaces, i.e.~spaces of the form $H^2 \ominus θH^2$ where $θ$ is inner. For these compressions of the shift operator, we provide a survey of results on the connection between their numerical ranges and the numerical ranges of their unitary dilations. We also discuss related results for compressed shift operators on the bidisk associated to rational inner functions and conclude the paper with a brief discussion of the Crouzeix conjecture.

math.FA↗

arXiv2018

A multiplier algebra functional calculus

This paper generalizes the classical Sz.-Nagy--Foias $H^{\infty}(\mathbb{D})$ functional calculus for Hilbert space contractions. In particular, we replace the single contraction $T$ with a tuple $T=(T_1, \dots, T_d)$ of commuting bounded operators on a Hilbert space and replace $H^{\infty}(\mathbb{D})$ with a large class of multiplier algebras of Hilbert function spaces on the unit ball in $\mathbb C^d$.

math.FA↗