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John E. McCarthy

Publications and source records attributed to John E. McCarthy.

At least 37 records · Page 2Linked to original sources
arXiv2020

Symbol functions for symmetric frameworks

We prove a variant of the well-known result that intertwiners for the bilateral shift on `$\ell^2(Z)$ are unitarily equivalent to multiplication operators on $L^2(T)$. This enables us to unify and extend fundamental aspects of rigidity theory for bar-joint frameworks with an abelian symmetry group. In particular, we formulate the symbol function for a wide class of frameworks and show how to construct generalised rigid unit modes in a variety of new contexts.

math.FA↗
arXiv2020

An $H^p$ scale for complete Pick spaces

We define by interpolation a scale analogous to the Hardy $H^p$ scale for complete Pick spaces, and establish some of the basic properties of the resulting spaces, which we call $\mathcal{H}^p$. In particular, we obtain an $\mathcal{H}^p-\mathcal{H}^q$ duality and establish sharp pointwise estimates for functions in $\mathcal{H}^p$.

math.FA↗
arXiv2020

Interpolating sequences in spaces with the complete Pick property

We characterize interpolating sequences for multiplier algebras of spaces with the complete Pick property. Specifically, we show that a sequence is interpolating if and only if it is separated and generates a Carleson measure. This generalizes results of Carleson for the Hardy space and of Bishop, Marshall and Sundberg for the Dirichlet space. Furthermore, we investigate interpolating sequences for pairs of Hilbert function spaces.

math.FA↗
arXiv2019

Calcular Algebras

A calcular algebra is a subalgebra of $H^\infty(Ω)$ with norm given by $\| ϕ\| = \sup \| ϕ(T) \|$ as $T$ ranges over a given class of commutative $d$-tuples of operators with Taylor spectrum in $Ø$. We discuss what algebras arise this way, and how they can be represented.

math.FA↗
arXiv2018

Factorizations induced by complete Nevanlinna-Pick factors

We prove a factorization theorem for reproducing kernel Hilbert spaces whose kernel has a normalized complete Nevanlinna-Pick factor. This result relates the functions in the original space to pointwise multipliers determined by the Nevanlinna-Pick kernel and has a number of interesting applications. For example, for a large class of spaces including Dirichlet and Drury-Arveson spaces, we construct for every function $f$ in the space a pluriharmonic majorant of $|f|^2$ with the property that whenever the majorant is bounded, the corresponding function $f$ is a pointwise multiplier.

math.FA↗
arXiv2018

Radially weighted Besov spaces and the Pick property

For $s\in \mathbb R$ the weighted Besov space on the unit ball $\mathbb B_d$ of $\mathbb C^d$ is defined by $B^s_ω=\{f\in \operatorname{Hol}(\mathbb B_d): \int_{\mathbb B_d}|R^sf|^2 ωdV<\infty\}.$ Here $R^s$ is a power of the radial derivative operator $R= \sum_{i=1}^d z_i\frac{\partial}{\partial z_i}$, $V$ denotes Lebesgue measure, and $ω$ is a radial weight function not supported on any ball of radius $< 1$. Our results imply that for all such weights $ω$ and $ν$, every bounded column multiplication operator $B^s_ω\to B^t_ν\otimes \ell^2$ induces a bounded row multiplier $B^s_ω\otimes \ell^2 \to B^t_ν$. Furthermore we show that if a weight $ω$ satisfies that for some $α>-1$ the ratio $ω(z)/(1-|z|^2)^α$ is nondecreasing for $t_0<|z|<1$, then $B^s_ω$ is a complete Pick space, whenever $s\ge (α+d)/2$.

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arXiv2018

A multiplier algebra functional calculus

This paper generalizes the classical Sz.-Nagy--Foias $H^{\infty}(\mathbb{D})$ functional calculus for Hilbert space contractions. In particular, we replace the single contraction $T$ with a tuple $T=(T_1, \dots, T_d)$ of commuting bounded operators on a Hilbert space and replace $H^{\infty}(\mathbb{D})$ with a large class of multiplier algebras of Hilbert function spaces on the unit ball in $\mathbb C^d$.

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arXiv2018

The Krzyż Conjecture and an Entropy Conjecture

We show that if the minimum entropy for a polynomial with roots on the unit circle is attained by polynomials with equally spaced roots, then, under a generic hypothesis about the nature of the extremum, the Krzyz conjecture on the maximum modulus of the Taylor coefficients of a holomorphic function that maps the disk to the punctured disk is true.

math.CV↗
arXiv2017

A non-commutative Julia Inequality

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 $\mathbb{C}^d$. We look at differentiability at a boundary point for functions that have a certain regularity there.

math.CV↗
arXiv2017

The Smirnov class for spaces with the complete Pick property

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.

math.FA↗
arXiv2016

The Julia-Caratheodory theorem on the bidisk revisited

The Julia quotient measures the ratio of the distance of a function value from the boundary to the distance from the boundary. The Julia-Carathéodory theorem on the bidisk states that if the Julia quotient is bounded along some sequence of nontangential approach to some point in the torus, the function must have directional derivatives in all directions pointing into the bidisk. The directional derivative, however, need not be a linear function of the direction in that case. In this note, we show that if the Julia quotient is uniformly bounded along every sequence of nontangential approach, the function must have a linear directional derivative. Additionally, we analyze a weaker condition, corresponding to being Lipschitz near the boundary, which implies the existence of a linear directional derivative for rational functions.

math.CV↗
arXiv2016

Nevanlinna-Pick Kernels and Localization

We describe those reproducing kernel Hilbert spaces of holomorphic functions on domains in ${\Bbb C}^d$ for which an analogue of the Nevanlinna-Pick theorem holds, in other words when the existence of a (possibly matrix-valued) function in the unit ball of the multiplier algebra with specified values on a finite set of points is equivalent to the positvity of a related matrix. Our description is in terms of a certain localization property of the kernel.

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