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- Free holomorphic functions (3)
- Hilbert spaces with reproducing kernels (3)
- Mathematics (3)
- Bidisk (2)
- Hardy space (2)
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- Maximal functions (2)
- NC functions (2)
- Operator theory (2)
- Polydisk (2)
- Weak factorization (2)
- $A_p$ spaces (1)
- $L^p$ spaces (1)
- Agler decomposition (1)
- Algebraic Geometry (1)
- Algebras (1)
- Analytic extension (1)
- Automorphisms (1)
- BMO space (1)
- Bmo(ℝ×ℝ) (1)
- Colligation (1)
- Commutator (1)
- Compactness (1)
- Complex Variables (1)
- Composition operator (1)
- Conjugate functions (1)
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- Corona theorems (1)
- Cyclicity (1)
- De Branges-Rovnyak spaces (1)
- Determinantal representations (1)
Articles 1 - 25 of 25
Full-Text Articles in Physical Sciences and Mathematics
Localization And Compactness Of Operators On Fock Spaces, Zhangjian Hu, Xiaofen Lv, Brett D. Wick
Localization And Compactness Of Operators On Fock Spaces, Zhangjian Hu, Xiaofen Lv, Brett D. Wick
Mathematics Faculty Publications
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A Non-Commutative Julia Inequality, John E. Mccarthy, James E. Pascoe
A Non-Commutative Julia Inequality, John E. Mccarthy, James E. Pascoe
Mathematics Faculty Publications
We prove a Julia inequality for bounded non-commutative functions on polynomial polyhedra. We use this to deduce a Julia inequality for holomorphic functions on classical domains in Cd. We look at differentiability at a boundary point for functions that have a certain regularity there.
Commutators, Little Bmo And Weak Factorization, Xuan Thinh Duong, Ji Li, Brett D. Wick, Dongyong Yang
Commutators, Little Bmo And Weak Factorization, Xuan Thinh Duong, Ji Li, Brett D. Wick, Dongyong Yang
Mathematics Faculty Publications
In this paper, we provide a direct and constructive proof of weak factorization of h1 (ℝ×ℝ) (the predual of little BMO space bmo(ℝ×ℝ) studied by Cotlar-Sadosky and Ferguson-Sadosky), i.e., for every f Є h1 (ℝ×ℝ) there exist sequences {αkj} Є l and functions gjk, hkj Є L2 (ℝ2 ) such that [Equation Unavailable] in the sense of h1 (ℝ×ℝ), where H1 and H2 are the Hilbert transforms on the first and second variable, respectively. Moreover, the norm ║fh1║(ℝ×ℝ) is given in terms of ║gjk║ L2(ℝ2) and ║hkj║ L2(ℝ2). By duality, this directly implies a lower bound on the norm of …
Smirnov Class For Spaces With The Complete Pick Property, Alexandru Aleman, Michael Hartz, John E. Mccarthy, Stefan Richter
Smirnov Class For Spaces With The Complete Pick Property, Alexandru Aleman, Michael Hartz, John E. Mccarthy, Stefan Richter
Mathematics Faculty Publications
We show that every function in a reproducing kernel Hilbert space with a normalized complete Pick kernel is the quotient of a multiplier and a cyclic multiplier. This extends a theorem of Alpay, Bolotnikov and Kaptanoğlu. We explore various consequences of this result regarding zero sets, spaces on compact sets and Gleason parts. In particular, using a construction of Salas, we exhibit a rotationally invariant complete Pick space of analytic functions on the unit disc for which the corona theorem fails.
Weak Factorizations Of The Hardy Space H1(RN) In Terms Of Multilinear Riesz Transforms, Ji Li, Brett D. Wick
Weak Factorizations Of The Hardy Space H1(RN) In Terms Of Multilinear Riesz Transforms, Ji Li, Brett D. Wick
Mathematics Faculty Publications
This paper provides a constructive proof of the weak factorization of the classical Hardy space in terms of multilinear Riesz transforms. As a direct application, we obtain a new proof of the characterization of (the dual of ) via commutators of the multilinear Riesz transforms.
Spaces Of Dirichlet Series With The Complete Pick Property, John E. Mccarthy, Orr Moshe Shalit
Spaces Of Dirichlet Series With The Complete Pick Property, John E. Mccarthy, Orr Moshe Shalit
Mathematics Faculty Publications
We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form k(s,u)=∑ann−s−u¯, and characterize when such a space is a complete Pick space. We then discuss what it means for two reproducing kernel Hilbert spaces to be “the same”, and introduce a notion of weak isomorphism. Many of the spaces we consider turn out to be weakly isomorphic as reproducing kernel Hilbert spaces to the Drury–Arveson space Hd2 in d variables, where d can be any number in {1, 2,...,∞}, and in particular their multiplier algebras are unitarily equivalent to the multiplier algebra of Hd2. Thus, a …
The Corona Problem For Kernel Multiplier Algebras, Eric T. Sawyer, Brett D. Wick
The Corona Problem For Kernel Multiplier Algebras, Eric T. Sawyer, Brett D. Wick
Mathematics Faculty Publications
We prove an alternate Toeplitz corona theorem for the algebras of pointwise kernel multipliers of Besov-Sobolev spaces on the unit ball in C, and for the algebra of bounded analytic functions on certain strictly pseudoconvex domains and polydiscs in higher dimensions as well. This alternate Toeplitz corona theorem extends to more general Hilbert function spaces where it does not require the complete Pick property. Instead, the kernel functions kx (y) of certain Hilbert function spaces H are assumed to be invertible multipliers on H and then we continue a research thread begun by Agler and McCarthy in 1999, and continued …
Non-Commutative Automorphisms Of Bounded Non-Commutative Domains, John E. Mccarthy, Richard M. Timoney
Non-Commutative Automorphisms Of Bounded Non-Commutative Domains, John E. Mccarthy, Richard M. Timoney
Mathematics Faculty Publications
We establish rigidity (or uniqueness) theorems for non-commutative (NC) automorphisms that are natural extensions of classical results of H. Cartan and are improvements of recent results. We apply our results to NC domains consisting of unit balls of rectangular matrices.
Determinantal Representations Of Semihyperbolic Polynomials, Greg Knese
Determinantal Representations Of Semihyperbolic Polynomials, Greg Knese
Mathematics Faculty Publications
We prove a generalization of the Hermitian version of the Helton–Vinnikov determinantal representation for hyperbolic polynomials to the class of semihyperbolic polynomials, a strictly larger class, as shown by an example. We also prove that certain hyperbolic polynomials affine in two out of four variables divide a determinantal polynomial. The proofs are based on work related to polynomials with no zeros on the bidisk and tridisk.
A Remark On The Multipliers On Spaces Of Weak Products Of Functions, Stefan Richter, Brett D. Wick
A Remark On The Multipliers On Spaces Of Weak Products Of Functions, Stefan Richter, Brett D. Wick
Mathematics Faculty Publications
Operators on function spaces of form Cɸf = f ∘ ɸ, where ɸ is a fixed map are called composition operators with symbol ɸ. We study such operators acting on the Hilbert Hardy space over the right half-plane and characterize the situations when they are invertible, Fredholm, unitary, and Hermitian. We determine the normal composition operators with inner, respectively with Möbius symbol. In select cases, we calculate their spectra, essential spectra, and numerical ranges.
Polynomials With No Zeros On A Face Of The Bidisk, Jeffrey S. Geronimo, Plamen Iliev, Greg Knese
Polynomials With No Zeros On A Face Of The Bidisk, Jeffrey S. Geronimo, Plamen Iliev, Greg Knese
Mathematics Faculty Publications
We present a Hilbert space geometric approach to the problem of characterizing the positive bivariate trigonometric polynomials that can be represented as the square of a two variable polynomial possessing a certain stability requirement, namely no zeros on a face of the bidisk. Two different characterizations are given using a Hilbert space structure naturally associated to the trigonometric polynomial; one is in terms of a certain orthogonal decomposition the Hilbert space must possess called the “split-shift orthogonality condition” and another is an operator theoretic or matrix condition closely related to an earlier characterization due to the first two authors. This …
The Implicit Function Theorem And Free Algebraic Sets, Jim Agler, John E. Mccarthy
The Implicit Function Theorem And Free Algebraic Sets, Jim Agler, John E. Mccarthy
Mathematics Faculty Publications
We prove an implicit function theorem for non-commutative functions. We use this to show that if p ( X;Y ) is a generic non-commuting polynomial in two variables, and X is a generic matrix, then all solutions Y of p ( X;Y ) = 0 will commute with X
Aspects Of Non-Commutative Function Theory, Jim Agler, John E. Mccarthy
Aspects Of Non-Commutative Function Theory, Jim Agler, John E. Mccarthy
Mathematics Faculty Publications
We discuss non commutative functions, which naturally arise when dealing with functions of more than one matrix variable.
Cyclic Polynomials In Two Variables, Catherine Bénéteau, Greg Knese, Łukasz Kosiński, Constanze Liaw, Daniel Seco, Alan Sola
Cyclic Polynomials In Two Variables, Catherine Bénéteau, Greg Knese, Łukasz Kosiński, Constanze Liaw, Daniel Seco, Alan Sola
Mathematics Faculty Publications
We give a complete characterization of polynomials in two complex variables that are cyclic with respect to the coordinate shifts acting on Dirichlet-type spaces in the bidisk, which include the Hardy space and the Dirichlet space of the bidisk. The cyclicity of a polynomial depends on both the size and nature of the zero set of the polynomial on the distinguished boundary. The techniques in the proof come from real analytic function theory, determinantal representations for polynomials, and harmonic analysis on curves.
Commutators In The Two-Weight Setting, Irina Holmes, Michael T. Lacey, Brett D. Wick
Commutators In The Two-Weight Setting, Irina Holmes, Michael T. Lacey, Brett D. Wick
Mathematics Faculty Publications
Let R be the vector of Riesz transforms on Rn and let μ,λ∈Ap be two weights on Rn, 1p(μ)→Lp(λ)|| is shown to be equivalent to the function b being in a BMO space adapted to μ and λ. This is a common extension of a result of Coifman–Rochberg–Weiss in the case of both λ and μ being Lebesgue measure, and Bloom in the case of dimension one.
Unions Of Lebesgue Spaces And A1 Majorants, Greg Knese, John E. Mccarthy, Kabe Moen
Unions Of Lebesgue Spaces And A1 Majorants, Greg Knese, John E. Mccarthy, Kabe Moen
Mathematics Faculty Publications
We study two questions. When does a function belong to the union of Lebesgue spaces, and when does a function have an A1 majorant? We provide a systematic study of these questions and show that they are fundamentally related. We show that the union ofLwp(ℝn)spaces withw∈Apis equal to the union of all Banach function spaces for which the Hardy–Littlewood maximal function is bounded on the space itself and its associate space.
The Von Neumann Inequality For 3 × 3 Matrices, Greg Knese
The Von Neumann Inequality For 3 × 3 Matrices, Greg Knese
Mathematics Faculty Publications
This note details how recent work of Kosiński on the three point Pick interpolation problem on the polydisc can be used to prove the von Neumann inequality for d-tuples of commuting 3 x 3 contractive matrices.
Canonical Agler Decompositions And Transfer Function Realizations, Kelly Bickel, Greg Knese
Canonical Agler Decompositions And Transfer Function Realizations, Kelly Bickel, Greg Knese
Mathematics Faculty Publications
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 …
Pick Interpolation For Free Holomorphic Functions, Jim Agler, John E. Mccarthy
Pick Interpolation For Free Holomorphic Functions, Jim Agler, John E. Mccarthy
Mathematics Faculty Publications
We give necessary and sufficient conditions to solve an interpolation problem for free holomorphic functions bounded in norm on a free polynomial polyhedron. As an application, we prove that every bounded holomorphic function on a polynomial polyhedron extends to a bounded free function.
Non-Commutative Holomorphic Functions On Operator Domains, Jim Agler, John E. Mccarthy
Non-Commutative Holomorphic Functions On Operator Domains, Jim Agler, John E. Mccarthy
Mathematics Faculty Publications
We characterize functions of d-tuples of bounded operators on a Hilbert space that are uniformly approximable by free polynomials on balanced open sets.
Integrability And Regularity Of Rational Functions, Greg Knese
Integrability And Regularity Of Rational Functions, Greg Knese
Mathematics Faculty Publications
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 functions is to fix the denominator and look at the ideal of polynomials in the numerator such that the rational function is square integrable. A concrete list of generators is given for this ideal as well as a precise count of the dimension of the subspace of numerators with a specified bound on bidegree. The dimension count is accomplished by constructing a natural pair of commuting …
Global Holomorphic Functions In Several Noncommuting Variables, Jim Agler, John E. Mccarthy
Global Holomorphic Functions In Several Noncommuting Variables, Jim Agler, John E. Mccarthy
Mathematics Faculty Publications
We define a free holomorphic function to be a function that is locally, with respect to the free topology, a bounded nc-function. We prove that free holomorphic functions are the functions that are locally uniformly approximable by free polynomials. We prove a realization formula and an Oka-Weil theorem for free analytic functions.
Thin Sequences And The Gram Matrix, Pamela Gorkin, John E. Mccarthy, Sandra Pott, Brett D. Wick
Thin Sequences And The Gram Matrix, Pamela Gorkin, John E. Mccarthy, Sandra Pott, Brett D. Wick
Mathematics Faculty Publications
We provide a new proof of Volberg’s Theorem characterizing thin interpolating sequences as those for which the Gram matrix associated to the normalized reproducing kernels is a compact perturbation of the identity. In the same paper, Volberg characterized sequences for which the Gram matrix is a compact perturbation of a unitary as well as those for which the Gram matrix is a Schatten-2 class perturbation of a unitary operator. We extend this characterization from 2 to p, where 2 p ≤∞.
Hankel Vector Moment Sequences And The Non-Tangential Regularity At Infinity Of Two Variable Pick Functions, Jim Agler, John E. Mccarthy
Hankel Vector Moment Sequences And The Non-Tangential Regularity At Infinity Of Two Variable Pick Functions, Jim Agler, John E. Mccarthy
Mathematics Faculty Publications
A Pick function of variables is a holomorphic map from to , where is the upper halfplane. Some Pick functions of one variable have an asymptotic expansion at infinity, a power series with real numbers that gives an asymptotic expansion on non-tangential approach regions to infinity. In 1921 H. Hamburger characterized which sequences can occur. We give an extension of Hamburger's results to Pick functions of two variables.
Operator Monotone Functions And Löwner Functions Of Several Variables, Jim Agler, John E. Mccarthy, N J. Young
Operator Monotone Functions And Löwner Functions Of Several Variables, Jim Agler, John E. Mccarthy, N J. Young
Mathematics Faculty Publications
We prove generalizations of Loewner's results on matrix monotone functions to several variables. We give a characterization of when a function of d variables is locally monotone on d-tuples of commuting self-adjoint n-by-n matrices. We prove a generalization to several variables of Nevanlinna's theorem describing analytic functions that map the upper half-plane to itself and satisfy a growth condition. We use this to characterize all rational functions of two variables that are operator monotone.