# Research

One can find a list of my professional publications on my **CV** but below is a summary of areas I currently find interesting.

**Range spaces of co-analytic Toeplitz operators**: This is a paper with Emmanuel Fricain (Lille) and Andreas Hartmann (Bordeaux) on the boundary behavior of functions belonging to the range of co-anaytic Toeplitz operators. In a way, we have a Ahern-Clark type of theorem for this range. In addition, we give an natural orthogonal decomposition of the ranges of these operators in a suitable range space topology.

**Book on model spaces:** Together with Stephan Garcia (Pomona College) and Javad Mashreghi (Laval University), I recently wrote a book *Introduction to Model spaces and their Operators* that will appear with Cambridge University Press. This book is written for graduate students and researchers needing a friendly introduction to the subject.

**What is in a deBranges-Rovnyak space?** These spaces are defined with via the range of a certain defect operator or by their reproducing kernel and their exact contents are mysterious. In this paper with Fricain and Hartmann, we characterize, for certain reasonable parameters, the exact contents of these spaces.

**Partial orders on partial isometries:** This is a paper with Stephan Garcia (Pomona) and Rob Martin (Cape Town, South Africa) which discusses various pre and partial orders one can put on the set of partial isometries on Hilbert spaces. This study was continued by Dan Timoton in a paper which generalized some of these results to pre and partial orders on contractions.

**Carleson measures for deBranges-Rovnyak spaces:** My work with Fricain and Hartmann on Carleson and reverse Carleson measures on various sub-Hardy Hilbert spaces continue with this paper.

**Extremal problems for characteristic functions**: In some recent work with Isabelle Chalendar, Stephan Garcia, and Dan Timotin, I examined the distance from from characteristic functions of sets in the unit circle (as well as z^n times these characteristic functions) to the algebra of bounded analytic functions on the open unit disk. This work relates to Hankel operators, Toeplitz operators, conformal mappings, and classical extremal problems of H. S. Shapiro and Dima Khavinson.

Extremal problems for characteristic functions (talk)

**Lectures on Model spaces**: This summer I gave a series of 8 lectures at the University of Helsinki on the basics of model spaces. Here are notes on these lectures (which are in the processing of becoming a book). As a warm up exercise for our book, Stephan Garcia and I wrote up a survey paper on model spaces.

**Lectures on Clark measures**: This summer I also gave a series of three lectures at a workshop at Lens University on the basics of Clark measures.

**Symmetric operators**: In recent years, I have been working with Alexandru Aleman and Rob Martin on projects dealing with unitarily equivalence of unbounded symmetric operators. It is well known that this problem has been settled by some old results of Livsic. In the paper below, we give a new proof of Livsic’s theorem (which does not involve the spectral theorem) and give some techniques on how to compute the Livsic characteristic. Our techniques involve reproducing kernel Hilbert spaces.

On a theorem of Livsic (talk on this paper)

**Model spaces**: It is well known that functions in the model spaces (the orthogonal complement) of I H^2, where I is inner, have some regularity at certain points on the unit circle. For example, it is well known that functions in the model space have analytic continuation at all points which avoid the spectrum of the inner function. It is also well known (Ahern and Clark) that every function in the model space has a finite non-tangential limit at a boundary point if and only if the inner function I has a finite angular derivative at that point. In the following two papers

Bad boundary behavior in star invariant subspaces I

Bad boundary behavior in star invariant subspaces II

with Andreas Hartmann, we explore the growth rates of functions in the model spaces where the angular derivative of the inner function is infinite.

In some recent work with Blandignères, Fricain, Gaunard, and Hartmann, we examine reverse Carleson embedding theorems for model spaces.

Reverse Carleson embeddings for model spaces

**The Jordan curve theorem**: I recently wrote a paper with my wife (who is an artist) on the beauty and complexity of the Jordan curve theorem through art. The paper is titled The Jordan curve theorem is non-trivial.

**Truncated Toeplitz operators: **This is a nice topic which has been interesting me (and others) for the past several years. It all started with paper of Don Sarason “Algebraic properties of truncated Toeplitz operators” where he laid out the basic properties of these operators and asked several questions. Here some of my recent papers (with Cima, Garcia, Hartmann, and Wogen) on TTOs.

C* algebras generated by truncated Toeplitz operators

Recent progress on truncated Toeplitz operators

Truncated Toeplitz operators on finite dimensional spaces

A non-linear extremal problem for the Hardy space

The norm of a truncated Toeplitz operator

Spatial isomorphisms of truncated Toeplitz opertors

Spatial isomorphisms of algebras of truncated Toeplitz operators (recent talk)

Unitary equivalence to a truncated Toeplitz operator

Truncated Toeplitz operators and nearly invariant subspaces

Boundary values in range spaces of truncated Toeplitz operators (recent talk) (another recent talk)

**Hardy spaces of slit domains: **Ever since my Ph.D. thesis, I’ve had an interest in invariant subspaces, under the shift operator, of various Banach spaces of analytic functions on slit domains. In a paper with Alexandru Aleman and Nathan Feldman, we characterized the invariant subspaces of Hardy spaces of slit domains. Our paper *The Hardy space of a slit domain* looks not only at the invariant subspaces but nearly invariant subspaces. This paper has appeared in book form as part of Birkhauser’s Frontiers in Mathematics series. I gave a talk on this paper at a complex analysis conference at UNC-Chapel Hill.

**Common cyclic vectors: **In two papers with Warren Wogen, we examine whether or not certain natural classes of multiplication operators (normal, unitary) on the Lebesgue spaces have common cyclic vectors.

**The Cauchy transform: **The Cauchy transform is a well-studied object in complex analysis – via the Cauchy integral formula. In a book *The Cauchy Transform* with A. Matheson and J. Cima, we gather up many results about the Cauchy transform on the circle including the famous classical theorems of Cauchy, Privaliv, Smirnov. Riesz, Kolmogorov together with more modern results of Khruschev, Vinogradov, Aleksandrov, and Poltoratski. To get started on this subject, we wrote an expository paper on this subject. I have been evangelizing recently about Cauchy transform in various taks.

**The backward shift operator: **Several years ago, I wrote book with J. Cima which thoroughly discusses the invariant subspaces of the backward shift operator (often called star-invariant) on the Hardy spaces Hp. For p > 1, this work is standard and covered in a seminar paper of Douglas, Shapiro, and Shields – later discussed in various forms in a well-known book of Nikolskii (A treatise on the shift operator). When p = 1, the characterization of the star-invariant subspaces is the same except the details are somewhat different. When p < 1, all hell breaks loose and the description of the star-invariant subspaces is different. Fortunately, A. Aleksandrov sorts this all out in a paper which, unfortunately to the West, was never translated into English. Our book, *The backward shift on the Hardy space* (link) provides a thorough treatment of this subject replete with all the details and (hopefully) accessible to an advanced graduate student in complex analysis. In case you don’t want to wade through a book in this subject, I have an expository paper on star-invariant subspaces.

**Generalized analytic continuation: **When can we say, in some reasonable way, that component functions of a meromorphic function on a disconnected domain, are “continuations” of each other? What role do such “continuations” play in certain aspects of approximation theory and operator theory? In a book Generalized Analytic Continuation with H. S. Shapiro, we discuss these questions and a whole host of other ideas from complex analysis. I’ll point you to two other papers (1) (2) and a recent talk which also deal with the topic of generalized generalized analytic continuation.

**The classical Dirichlet space: **I’ve always been interested in the classical Dirichlet space of analytic functions on the unit disk whose derivative is square integrable with respect to area measure. The Dirichlet space is a fascinating Hilbert space of analytic functions and I have written a survey paper on this subject.