Related papers: Inner functions on the bidisk and associated Hilbe…
We use shift-invariant subspaces of the Hardy space on the bidisk to provide an elementary proof of the Agler Decomposition Theorem. We observe that these shift-invariant subspaces are specific cases of Hilbert spaces that can be defined…
Every two variable rational inner function on the bidisk has a special representation called a transfer function realization. It is well known and related to important ideas in operator theory that this does not extend to three or more…
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…
We study representations of inner functions on the bidisc from a fractional linear transformation point of view, and provide sufficient conditions, in terms of colligation matrices, for the existence of two-variable inner functions. Here…
Motivated by recent work in the mathematics and engineering literature, we study integrability and non-tangential regularity on the two-torus for rational functions that are holomorphic on the bidisk. One way to study such rational…
We analyze the singularities of rational inner functions on the unit bidisk and study both when these functions belong to Dirichlet-type spaces and when their partial derivatives belong to Hardy spaces. We characterize derivative…
We discuss the notion of an inner function for spaces of analytic functions in multiply connected domains in $\mathbb{C}$, giving a historical overview and comparing several possible definitions. We explore connections between inner…
We give a simplified exposition of Kummert's approach to proving that every matrix-valued rational inner function in two variables has a minimal unitary transfer function realization. A slight modification of the approach extends to…
A natural connection between rational functions of several real or complex variables, and subspace collections is explored. A new class of function, superfunctions, are introduced which are the counterpart to functions at the level of…
Rational inner functions are a generalization of finite Blaschke products to several variables. In this article we survey a variety of results about rational inner functions related to interpolation, sums of squares formulas, and boundary…
Inner functions play a central role in function theory and operator theory on the Hardy space over the unit disk. Motivated by recent works of C. B\'en\'eteau et al. and of D. Seco, we discuss inner functions on more general weighted Hardy…
We return to Takagi's variational principle, generalized after forty years to two complex variables by Pfister. Both isolating some extremal rational functions associated to a bounded holomorphic function in the unit disk, respectively the…
We study membership of rational inner functions on the bidisk $\mathbb{D}^2$ in a scale of Dirichlet spaces considered by Bera, Chavan, and Ghara, and in higher-order variants of these spaces. We give a characterization for membership in…
We discuss various theorems about bounded analytic functions on the bidisk that were proved using operator theory.
In previous work, the authors studied the compressed shift operators $S_{z_1}$ and $S_{z_2}$ on two-variable model spaces $H^2(\mathbb{D}^2)\ominus \theta H^2(\mathbb{D}^2)$, where $\theta$ is a two-variable scalar inner function. Among…
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…
A seminal result of Agler proves that the natural de Branges-Rovnyak kernel function associated to a bounded analytic function on the bidisk can be decomposed into two shift-invariant pieces. Agler's decomposition is non-constructive-a…
For $1/2<p<1$, a description of inner functions whose derivative is in the Hardy space $H^p$ is given in terms of either their mapping properties or the geometric distribution of their zeros.
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…
We single out some problems of Schubert calculus of subspaces of codimension 2 that have the property that all their solutions are real provided that the data are real. Our arguments explore the connection between subspaces of codimension 2…