The Wedge-of-the-edge Theorem: Edge-of-the-wedge Type Phenomenon Within the Common Real Boundary

Pascoe, J. E.

doi:10.4153/cmb-2018-025-3

Cited by 7 publications

(5 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then by Theorem 2.1 in [Pas19] there is an open set D ⊆ C d only depending on W so that f n analytically continues to D. Letting x n get sufficiently close to 0, this implies that f analytically continues to 0, a contradiction. Thus T d ∩ Z p ⊆ C λ , and the result follows.…”

Section: General Rifsmentioning

confidence: 95%

Singularities of rational inner functions in higher dimensions

Bickel¹,

Pascoe²,

Sola³

2019

Preprint

View full text Add to dashboard Cite

We study the boundary behavior of rational inner functions (RIFs) in dimensions three and higher from both analytic and geometric viewpoints. On the analytic side, we use the critical integrability of the derivative of a rational inner function of several variables to quantify the behavior of a RIF near its singularities, and on the geometric side we show that the unimodular level sets of a RIF convey information about its set of singularities. We then specialize to three-variable degree (m, n, 1) RIFs and conduct a detailed study of their derivative integrability, zero set and unimodular level set behavior, and non-tangential boundary values. Our results, coupled with constructions of non-trivial RIF examples, demonstrate that much of the nice behavior seen in the two-variable case is lost in higher dimensions.

show abstract

Section: General Rifsmentioning

confidence: 95%

Singularities of rational inner functions in higher dimensions

Bickel¹,

Pascoe²,

Sola³

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…The edge-of-the-wedge theorem (proven by Bogoliubov and treated by Rudin in a series of lectures [41]) is useful in showing that such a continuation exists. Extremely flexible generalizations of this result to several variables have appeared in [36,34]. The key lemma from [36] follows, which we will need to generate quantitative bounds.…”

Section: Prelude: the Quantitative Wedge-of-the-edge Theoremmentioning

confidence: 95%

“…Extremely flexible generalizations of this result to several variables have appeared in [36,34]. The key lemma from [36] follows, which we will need to generate quantitative bounds. In this section, we prove a version of the wedge-of-the-edge theorem in the operator system setting.…”

Section: Prelude: the Quantitative Wedge-of-the-edge Theoremmentioning

confidence: 95%

See 1 more Smart Citation

The royal road to automatic noncommutative real analyticity, monotonicity, and convexity

Pascoe¹,

Tully‐Doyle²

2019

Preprint

Self Cite

View full text Add to dashboard Cite

It was shown classically that matrix monotone and matrix convex functions must be real analytic by Löwner and Kraus respectively. Recently, various analogues have been found in several noncommuting variables. We develop a general framework for lifting automatic analyticity theorems in matrix analysis from one variable to several variables, the so-called "royal road theorem." That is, we establish the principle that the hard part of proving any automatic analyticity theorem lies in proving the one variable theorem. We use our main result to prove the noncommutative Löwner and Kraus theorems over operator systems as examples, including an analogue of the "butterfly realization" of Helton-McCullough-Vinnikov for general analytic functions.

show abstract

“…We will need the following quantitative wedge-of-the-edge theorem, which appeared as Corollary 2.3 in [9], but is essentially previous work in [8,6].…”

Section: This Shows Thatmentioning

confidence: 99%

Automatic real analyticity and a regal proof of a commutative multivariate Löwner theorem

Pascoe¹,

Tully‐Doyle²

2019

Preprint

Self Cite

View full text Add to dashboard Cite

We adapt the "royal road" method used to simplify automatic analyticity theorems in noncommutative function theory to several complex variables. We show that certain families of functions must be real analytic if they have certain nice properties on one dimensional slices.) is a matrix monotone function of t whenever t ∈ (0, 1), the ϕ i are automorphisms of the upper half plane, and the tuple (ϕ 1 (t), . . . , ϕ d (t)) maps (0, 1) into E. We use the "royal road" to show that a function is matrix monotone lite if and only if it analytically continues to the multi-variate upper half plane as a map into the upper half plane. Moreover, matrix monotone lite functions in two variables are locally matrix monotone in the sense of Agler-McCarthy-Young.

show abstract

The Wedge-of-the-edge Theorem: Edge-of-the-wedge Type Phenomenon Within the Common Real Boundary

Cited by 7 publications

References 12 publications

Singularities of rational inner functions in higher dimensions

Singularities of rational inner functions in higher dimensions

The royal road to automatic noncommutative real analyticity, monotonicity, and convexity

Automatic real analyticity and a regal proof of a commutative multivariate Löwner theorem

Contact Info

Product

Resources

About