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William T. Ross

Publications and source records attributed to William T. Ross.

At least 37 records · Page 2Linked to original sources
arXiv2018

Inner functions and zero sets for $\ell^{p}_{A}$

In this paper we characterize the zero sets of functions from $\ell^{p}_{A}$ (the analytic functions on the open unit disk $D$ whose Taylor coefficients form an $\ell^p$ sequence) by developing a concept of an "inner function" modeled by Beurling's discussion of the Hilbert space $\ell^{2}_{A}$ (the classical Hardy space). The zero set criterion is used to construct families of zero sets which are not covered by classical results. In particular, it is proved that when $p > 2$, there are zero sets for $\ell^{p}_{A}$ which are not Blaschke sequences.

math.CV↗
arXiv2018

Optimal Weak Parallelogram Constants for $L^p$ Spaces

Inspired by Clarkson's inequalities for $L^p$ and continuing work from \cite{CR}, this paper computes the optimal constant $C$ in the weak parallelogram laws $$ \|f + g \|^r + C\|f - g\|^r \leq 2^{r-1}\big( \|f\|^r + \|g\|^r \big), $$ $$ \|f + g \|^r + C\|f- g \|^r \geq 2^{r-1}\big( \|f\|^r + \|g \|^r \big)$$ for the $L^p$ spaces, $1 < p < \infty$.

math.FA↗
arXiv2017

The range and valence of a real Smirnov function

We give a complete description of the possible ranges of real Smirnov functions (quotients of two bounded analytic functions on the open unit disk where the denominator is outer and such that the radial boundary values are real almost everywhere on the unit circle). Our techniques use the theory of unbounded symmetric Toeplitz operators, some general theory of unbounded symmetric operators, classical Hardy spaces, and an application of the uniformization theorem. In addition, we completely characterize the possible valences for these real Smirnov functions when the valence is finite. To do so we construct Riemann surfaces we call disk trees by welding together copies of the unit disk and its complement in the Riemann sphere. We also make use of certain trees we call valence trees that mirror the structure of disk trees.

math.CV↗
arXiv2016

Finite Blaschke products: a survey

A finite Blaschke product is a product of finitely many automorphisms of the unit disk. This brief survey covers some of the main topics in the area, including characterizations of finite Blaschke products, approximation theorems, derivatives and residues of finite Blaschke products, geometric localization of zeros, and selected other topics.

math.CV↗
arXiv2015

Real complex functions

We survey a few classes of analytic functions on the disk that have real boundary values almost everywhere on the unit circle. We explore some of their properties, various decompositions, and some connections these functions make to operator theory.

math.CV↗
arXiv2015

Range Spaces of Co-analytic Toeplitz Operators

We discuss the range spaces of Toeplitz operators with co-analytic symbols where we focus on the boundary behavior of the functions in these spaces as well as a natural orthogonal decomposition of this range.

math.CV↗
arXiv2015

Partial orders on partial isometries

This paper studies three natural pre-orders of increasing generality on the set of all completely non-unitary partial isometries with equal defect indices. We show that the problem of determining when one partial isometry is less than another with respect to these pre-orders is equivalent to the existence of a bounded (or isometric) multiplier between two natural reproducing kernel Hilbert spaces of analytic functions. For large classes of partial isometries these spaces can be realized as the well-known model subspaces and deBranges-Rovnyak spaces. This characterization is applied to investigate properties of these pre-orders and the equivalence classes they generate.

math.FA↗
arXiv2014

Concrete examples of $\mathscr{H}(b)$ spaces

In this paper we give an explicit description of de Branges-Rovnyak spaces $\HH(b)$ when $b$ is of the form $q^{r}$, where $q$ is a rational outer function in the closed unit ball of $H^{\infty}$ and $r$ is a positive number.

math.CV↗
arXiv2014

An extremal problem for characteristic functions

Suppose $E$ is a subset of the unit circle $\mathbb{T}$ and $H^\infty\subset L^\infty$ is the Hardy subalgebra. We examine the problem of finding the distance from the characteristic function of $E$ to $z^nH^\infty$. This admits an alternate description as a dual extremal problem. Precise solutions are given in several important cases. The techniques used involve the theory of Toeplitz and Hankel operators as well as the construction of certain conformal mappings.

math.CV↗
arXiv2013

Model spaces: a survey

This is a brief and gentle introduction, aimed at graduate students, to the subject of model subspaces of the Hardy space.

math.CV↗
arXiv2012

Reverse Carleson Embeddings for Model Spaces

The classical embedding theorem of Carleson deals with finite positive Borel measures $μ$ on the closed unit disk for which there exists a positive constant $c$ such that $|f|_{L^2(μ)} \leq c |f|_{H^2}$ for all $f \in H^2$, the Hardy space of the unit disk. Lefévre et al. examined measures $μ$ for which there exists a positive constant $c$ such that $\|f\|_{L^2(μ)} \geq c |f|_{H^2}$ for all $f \in H^2$. The first type of inequality above was explored with $H^2$ replaced by one of the model spaces $(ΘH^2)^{\perp}$ by Aleksandrov, Baranov, Cohn, Treil, and Volberg. In this paper we discuss the second type of inequality in $(ΘH^2)^{\perp}$.

math.CV↗
arXiv2012

Recent progress on truncated Toeplitz operators

This paper is a survey on the emerging theory of truncated Toeplitz operators. We begin with a brief introduction to the subject and then highlight the many recent developments in the field since Sarason's seminal paper in 2007.

math.CV↗
arXiv2012

$C^*$-algebras generated by truncated Toeplitz operators

We obtain an analogue of Coburn's description of the Toeplitz algebra in the setting of truncated Toeplitz operators. As a byproduct, we provide several examples of complex symmetric operators which are not unitarily equivalent to truncated Toeplitz operators having continuous symbols.

math.OA↗
arXiv2012

Bad boundary behavior in star invariant subspaces I

We discuss the boundary behavior of functions in backward shift invariant subspaces $(B H^2)^{\perp}$, where $B$ is a Blaschke product. Extending some results of Ahern and Clark, we are particularly interested in the growth rates of functions at points of the spectrum of $B$ where $B$ does not admit a derivative in the sense of Carathéodory.

math.CV↗