arXiv subjects

Kelly Bickel

Publications and source records attributed to Kelly Bickel.

33 records · Page 2Linked to original sources

arXiv2018

Level curve portraits of rational inner functions

We analyze the behavior of rational inner functions on the unit bidisk near singularities on the distinguished boundary $\mathbb{T}^2$ using level sets. We show that the unimodular level sets of a rational inner function $ϕ$ can be parametrized with analytic curves and connect the behavior of these analytic curves to that of the zero set of $ϕ$. We apply these results to obtain a detailed description of the fine numerical stability of $ϕ$: for instance, we show that $\frac{\partial ϕ}{\partial z_1}$ and $\frac{\partial ϕ}{\partial z_2}$ always possess the same $L^{\mathfrak{p}}$-integrability on $\mathbb{T}^2$, and we obtain combinatorial relations between intersection multiplicities at singularities and vanishing orders for branches of level sets. We also present several new methods of constructing rational inner functions that allow us to prescribe properties of their zero sets, unimodular level sets, and singularities.

math.CV↗

arXiv2017

Derivatives of rational inner functions: geometry of singularities and integrability at the boundary

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 $H^{\mathfrak{p}}$ membership purely in terms of contact order, a measure of the rate at which the zero set of a rational inner function approaches the distinguished boundary of the bidisk. We also show that derivatives of rational inner functions with singularities fail to be in $H^{\mathfrak{p}}$ for $\mathfrak{p}\ge\frac{3}{2}$ and that higher non-tangential regularity of a rational inner function paradoxically reduces the $H^{\mathfrak{p}}$ integrability of its derivative. We derive inclusion results for Dirichlet-type spaces from derivative inclusion for $H^{\mathfrak{p}}$. Using Agler decompositions and local Dirichlet integrals, we further prove that a restricted class of rational inner functions fails to belong to the unweighted Dirichlet space.

math.CV↗

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↗

arXiv2016

Two weight estimates with matrix measures for well localized operators

In this paper, we give necessary and sufficient conditions for weighted $L^2$ estimates with matrix-valued measures of well localized operators. Namely, we seek estimates of the form: \[ \| T(\mathbf{W} f)\|_{L^2(\mathbf{V})} \le C\|f\|_{L^2(\mathbf{W})} \] where $T$ is formally an integral operator with additional structure, $\mathbf{W}, \mathbf{V}$ are matrix measures, and the underlying measure space possesses a filtration. The characterization we obtain is of Sawyer-type; in particular we show that certain natural testing conditions obtained by studying the operator and its adjoint on indicator functions suffice to determine boundedness. Working in both the matrix weighted setting and the setting of measure spaces with arbitrary filtrations requires novel modifications of a T1 proof strategy; a particular benefit of this level of generality is that we obtain polynomial estimates on the complexity of certain Haar shift operators.

math.FA↗

arXiv2016

Characterizations of $A_2$ Matrix Power Weights

In the scalar setting, the power functions $|x|^γ$, for $-1 < γ<1$, are the canonical examples of $A_2$ weights. In this paper, we study two types of power functions in the matrix setting, with the goal of obtaining canonical examples of $A_2$ matrix weights. We first study Type 1 matrix power functions, which are $n\times n$ matrix functions whose entries are of the form $a|x|^γ.$ Our main result characterizes when these power functions are $A_2$ matrix weights and has two extensions to Type $1$ power functions of several variables. We also study Type 2 matrix power functions, which are $n\times n$ matrix functions whose eigenvalues are of the form $a|x|^γ.$ We find necessary conditions for these to be $A_2$ matrix weights and give an example showing that even nice functions of this form can fail to be $A_2$ matrix weights.

math.CA↗

arXiv2015

Bounds for the Hilbert Transform with Matrix $A_2$ Weights

Let $W$ denote a matrix $A_2$ weight. In this paper, we implement a scalar argument using the square function to deduce square-function type results for vector-valued functions in $L^2(\mathbb{R},\mathbb{C}^d)$. These results are then used to study the boundedness of the Hilbert transform and Haar multipliers on $L^2(\mathbb{R},\mathbb{C}^d)$. Our proof shortens the original argument by Treil and Volberg and improves the dependence on the $A_2$ characteristic. In particular, we prove that the Hilbert transform and Haar multipliers map $L^2(\mathbb{R},W,\mathbb{C}^d)$ to itself with dependence on on the $A_2$ characteristic at most $[W]_{A_2}^{\frac{3}{2}} \log [W]_{A_2}$.

math.CA↗

arXiv2015

Well-Localized Operators on Matrix Weighted $L^2$ Spaces

Nazarov-Treil-Volberg recently proved an elegant two-weight T1 theorem for "almost diagonal" operators that played a key role in the proof of the $A_2$ conjecture for dyadic shifts and related operators. In this paper, we obtain a generalization of their T1 theorem to the setting of matrix weights. Our theorem does differ slightly from the scalar results, a fact attributable almost completely to differences between the scalar and matrix Carleson Embedding Theorems. The main tools include a reduction to the study of well-localized operators, a new system of Haar functions adapted to matrix weights, and a matrix Carleson Embedding Theorem.

math.CA↗

arXiv2015

Properties of vector-valued submodules on the bidisk

In previous work, the authors studied the compressed shift operators $S_{z_1}$ and $S_{z_2}$ on two-variable model spaces $H^2(\mathbb{D}^2)\ominus θH^2(\mathbb{D}^2)$, where $θ$ is a two-variable scalar inner function. Among other results, the authors used Agler decompositions to characterize the ranks of the operators $[S_{z_j}, S^*_{z_j}]$ in terms of the degree of rational $θ.$ In this paper, we examine similar questions for $H^2(\mathbb{D}^2)\ominus ΘH^2(\mathbb{D}^2)$ when $Θ$ is a matrix-valued inner function. We extend several results our previous work connecting $\text{Rank} [S_{z_j}, S^*_{z_j}]$ and the degree of $Θ$ to the matrix setting. When results do not clearly generalize, we conjecture what is true and provide supporting examples.

math.CV↗

arXiv2015

A Study of the Matrix Carleson Embedding Theorem with Applications to Sparse Operators

In this paper, we study the dyadic Carleson Embedding Theorem in the matrix weighted setting. We provide two new proofs of this theorem, which highlight connections between the matrix Carleson Embedding Theorem and both maximal functions and $H^1$-BMO duality. Along the way, we establish boundedness results about new maximal functions associated to matrix $A_2$ weights and duality results concerning $H^1$ and BMO sequence spaces in the matrix setting. As an application, we then use this Carleson Embedding Theorem to show that if $S$ is a sparse operator, then the operator norm of $S$ on $L^2(W)$ satisfies: \[ \| S\|_{L^2(W) \rightarrow L^2(W)} \lesssim [W]_{A_2}^{\frac{3}{2}},\] for every matrix $A_2$ weight $W$.

math.CA↗

arXiv2014

Properties of Beurling-Type Submodules via Agler Decompositions

In this paper, we study operator-theoretic properties of the compressed shift operators $S_{z_1}$ and $S_{z_2}$ on complements of submodules of the Hardy space over the bidisk $H^2(\mathbb{D}^2)$. Specifically, we study Beurling-type submodules - namely submodules of the form $θH^2(\mathbb{D}^2)$ for $θ$ inner - using properties of Agler decompositions of $θ$ to deduce properties of $S_{z_1}$ and $S_{z_2}$ on model spaces $H^2(\mathbb{D}^2) \ominus θH^2(\mathbb{D}^2)$. Results include characterizations (in terms of $θ$) of when a commutator $[S_{z_j}^*, S_{z_j}]$ has rank $n$ and when subspaces associated to Agler decompositions are reducing for $S_{z_1}$ and $S_{z_2}$. We include several open questions.

math.CV↗

arXiv2014

The Linear Bound for Haar Multiplier Paraproducts

We study the natural resolution of the conjugated Haar multiplier $M_{w^{\frac{1}{2}}}T_σM_{w^{-\frac{1}{2}}},$ where the multiplication operators $M_{w^{\pm\frac{1}{2}}}$ are decomposed into their canonical paraproduct decompositions. We prove that each constituent operator obtained from this resolution has a linear bound on $L^2(\mathbb{R}^d;w)$ in terms of the $A_{2}$ characteristic of $w$. The main tools used are a product formula for Haar coefficients, the Carleson Embedding Theorem, the linear bound for the square function, and the well-known linear bound of $T_σ$ on $L^2(\mathbb{R}^d,w).$

math.CA↗

arXiv2013

Canonical Agler Decompositions and Transfer Function Realizations

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 problem remedied by work of Ball-Sadosky-Vinnikov, which uses scattering systems to produce Agler decompositions through concrete Hilbert space geometry. This method, while constructive, so far has not revealed the rich structure shown to be present for special classes of functions-inner and rational inner functions. In this paper, we show that most of the important structure present in these special cases extends to general bounded analytic functions. We give characterizations of all Agler decompositions, we prove the existence of coisometric transfer function realizations with natural state spaces, and we characterize when Schur functions on the bidisk possess analytic extensions past the boundary in terms of associated Hilbert spaces.

math.FA↗

arXiv2012

Fundamental Agler Decompositions

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 from Agler decompositions and analyze the properties of such Hilbert spaces. We then restrict attention to rational inner functions and show that the shift-invariant subspaces provide easy proofs of several known results about decompositions of rational inner functions. We use our analysis to obtain a result about stable polynomials on the polydisk.

math.FA↗

arXiv2012

Inner functions on the bidisk and associated Hilbert spaces

Matrix valued inner functions on the bidisk have a number of natural subspaces of the Hardy space on the torus associated to them. We study their relationship to Agler decompositions, regularity up to the boundary, and restriction maps into one variable spaces. We give a complete description of the important spaces associated to matrix rational inner functions. The dimension of some of these spaces can be computed in a straightforward way, and this ends up having an application to the study of three variable rational inner functions. Examples are included to highlight the differences between the scalar and matrix cases.

math.FA↗

arXiv2012

Differentiating Matrix Functions

Multivariable, real-valued functions induce matrix-valued functions defined on the space of d-tuples of n-times-n pairwise-commuting self-adjoint matrices. We examine the geometry of this space of matrices and conclude that the best notion of differentiation of these matrix-valued functions is differentiation along curves. We prove that a C^1 real-valued function always induces a C^1 matrix function and give an explicit formula for the derivative. We also show that every real-valued C^m function defined on an open rectangle in the plane induces a matrix-valued function that can be m-times continuously differentiated along C^m curves.

math.FA↗