One can find a list of my professional publications on my CV but below is a summary of areas I currently find interesting.
Truncated Toeplitz operators
My most recent research interests are in truncated Toeplitz operators where I have a recent paper with Warren Wogen and Joseph Cima - Truncated Toeplitz operators on finite dimensional spaces (to appear in Operators and Matrices) - see here for a recent talk I presented. These operators were explored by Don Sarason in a recent paper Algebraic properties of truncated Toeplitz operators (published in Operators and Matrices)
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 recent paper with Alexandru Aleman and Nathan Feldman, we characterize 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. I recently gave a talk on this paper at a complex analysis conference at UNC-Chapel Hill.
Common cyclic vectors
In two relatively recent 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 recent 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.
Some recent presentations: