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Pamela Gorkin

Publications and source records attributed to Pamela Gorkin.

At least 19 recordsLinked to original sources
arXiv2022

Decomposable Blaschke products of degree $2^n$

We study the decomposability of a finite Blaschke product $B$ of degree $2^n$ into $n$ degree-$2$ Blaschke products, examining the connections between Blaschke products, the elliptical range theorem, Poncelet theorem, and the monodromy group. We show that if the numerical range of the compression of the shift operator, $W(S_B)$, with $B$ a Blaschke product of degree $n$, is an ellipse then $B$ can be written as a composition of lower-degree Blaschke products that correspond to a factorization of the integer $n$. We also show that a Blaschke product of degree $2^n$ with an elliptical Blaschke curve has at most $n$ distinct critical values, and we use this to examine the monodromy group associated with a regularized Blaschke product $B$. We prove that if $B$ can be decomposed into $n$ degree-$2$ Blaschke products, then the monodromy group associated with $B$ is the wreath product of $n$ cyclic groups of order $2$. Lastly, we study the group of invariants of a Blaschke product $B$ of order $2^n$ when $B$ is a composition of $n$ Blaschke products of order $2$.

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↗
arXiv2020

Interpolation in Model Spaces

In this paper we consider interpolation in model spaces, $H^2 \ominus B H^2$ with $B$ a Blaschke product. We study unions of interpolating sequences for two sequences that are far from each other in the pseudohyperbolic metric as well as two sequences that are close to each other in the pseudohyperbolic metric. The paper concludes with a discussion of the behavior of Frostman sequences under perturbations.

math.CV↗
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

Norms of truncated Toeplitz operators and numerical radii of restricted shifts

This paper gives a new approach to the calculation of the numerical radius of a restricted shift operator by linking it to the norm of a truncated Toeplitz operator (TTO), which can be be calculated by various methods. Further results on the norm of a TTO are derived, and a conjecture on the existence of continuous symbols for compact TTO is resolved.

math.FA↗
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↗
arXiv2017

Compressions of the Shift on the Bidisk and their Numerical Ranges

We consider two-variable model spaces associated to rational inner functions $Θ$ on the bidisk, which always possess canonical $z_2$-invariant subspaces $\mathcal{S}_2.$ A particularly interesting compression of the shift is the compression of multiplication by $z_1$ to $\mathcal{S}_2$, namely $ S^1_Θ:= P_{\mathcal{S}_2} M_{z_1} |_{\mathcal{S}_2}$. We show that these compressed shifts are unitarily equivalent to matrix-valued Toeplitz operators with well-behaved symbols and characterize their numerical ranges and radii. We later specialize to particularly simple rational inner functions and study the geometry of the associated numerical ranges, find formulas for the boundaries, answer the zero inclusion question, and determine whether the numerical ranges are ever circular.

math.CV↗
arXiv2015

Thin Sequences

We look at thin interpolating sequences and the role they play in uniform algebras, Hardy spaces, and model spaces.

math.CV↗
arXiv2015

Thin Sequences and Their Role in Model Spaces and Douglas Algebras

We study thin interpolating sequences $\{λ_n\}$ and their relationship to interpolation in the Hardy space $H^2$ and the model spaces $K_Θ= H^2 \ominus ΘH^2$, where $Θ$ is an inner function. Our results, phrased in terms of the functions that do the interpolation as well as Carleson measures, show that under the assumption that $Θ(λ_n) \to 0$ the interpolation properties in $H^2$ are essentially the same as those in $K_Θ$.

math.CV↗
arXiv2015

Inner functions and operator theory

This tutorial paper presents a survey of results, both classical and new, linking inner functions and operator theory. Topics discussed include invariant subspaces, universal operators, Hankel and Toeplitz operators, model spaces, truncated Toeplitz operators, restricted shifts, numerical ranges, and interpolation.

math.FA↗
arXiv2014

Thin sequences and the Gram matrix

We provide a new proof of Volberg's Theorem characterizing thin interpolating sequences as those for which the Gram matrix associated to the normalized reproducing kernels is a compact perturbation of the identity. In the same paper, Volberg characterized sequences for which the Gram matrix is a compact perturbation of a unitary as well as those for which the Gram matrix is a Schatten-$2$ class perturbation of a unitary operator. We extend this characterization from $2$ to $p$, where $2 \le p \le \infty$.

math.CV↗
arXiv2011

Orbits of non-elliptic disc automorphisms

Motivated by the Invariant Subspace Problem, we describe explicitly the closed subspace $H^2$ generated by the limit points in the $H^2$ norm of the orbit of a thin Blaschke product $B$ under composition operators $C_ϕ$ induced by non-elliptic automorphisms. This description exhibits a surprising connection to model spaces. Finally, we give a constructive characterization of the $C_ϕ$-eigenfunctions in $H^p$ for $1\le p\le \infty$.

math.FA↗
arXiv2011

The group of invariants of an inner function with finite spectrum

This paper determines the group of continuous invariants corresponding to an inner function $ Θ$ with finitely many singularities on the unit circle $T$; that is, the continuous mappings $g: T \to T$ such that $Θ\circ g = Θ$ on $\T$. These mappings form a group under composition.

math.CV↗
arXiv2010

Numerical ranges of $C_0(N)$ contractions

A conjecture of Halmos proved by Choi and Li states that the closure of the numerical range of a contraction on a Hilbert space is the intersection of the closure of the numerical ranges of all its unitary dilations. We show that for $C_0(N)$ contractions one can restrict the intersection to a smaller family of dilations. This generalizes a finite dimensional result of Gau and Wu.

math.FA↗
arXiv2010

Pervasive Algebras and Maximal Subalgebras

A uniform algebra $A$ on its Shilov boundary $X$ is {\em maximal} if $A$ is not $C(X)$ and there is no uniform algebra properly contained between $A$ and $C(X)$. It is {\em essentially pervasive} if $A$ is dense in $C(F)$ whenever $F$ is a proper closed subset of the essential set of $A$. If $A$ is maximal, then it is essentially pervasive and proper. We explore the gap between these two concepts. We show the following: (1) If $A$ is pervasive and proper, and has a nonconstant unimodular element, then $A$ contains an infinite descending chain of pervasive subalgebras on $X$. (2) It is possible to imbed a copy of the lattice of all subsets of $\N$ into the family of pervasive subalgebras of some $C(X)$. (3) In the other direction, if $A$ is strongly logmodular, proper and pervasive, then it is maximal. (4) This fails if the word \lq strongly' is removed. We discuss further examples, involving Dirichlet algebras, $A(U)$ algebras, Douglas algebras, and subalgebras of $H^\infty(\mathbb{D})$. We develop some new results that relate pervasiveness, maximality and relative maximality to support sets of representing measures.

math.FA↗
arXiv2010

Approximation by polynomials and Blaschke products having all zeros on a circle

We show that a nonvanishing analytic function on a domain in the unit disc can be approximated by (a scalar multiple of) a Blaschke product whose zeros lie on a prescribed circle enclosing the domain. We also give a new proof of the analogous classical result for polynomials. A connection is made to universality results for the Riemann zeta function.

math.CV↗