Möbius transformations and Blaschke products: The geometric connection

Daepp, Ulrich; Gorkin, Pamela; Shaffer, Andrew; Voss, Karl

doi:10.1016/j.laa.2016.11.032

Cited by 7 publications

(5 citation statements)

References 21 publications

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“…Then our computations require us to solve zB(z) = ±1. From (10) we see that the real parts of the roots satisfy…”

Section: Degree-3 Blaschke Productsmentioning

confidence: 99%

“…In this paper we focus on a particular class of operators known as compressions of the shift operator and consider the numerical radius of the operators in this class, where little is known. Much more is known about the geometry of the numerical range of these operators and we refer the reader to [10,23,24,25,26,35]. Though the work in these papers may also be used to solve some of the problems mentioned here, our goal is to add other methods that may be useful.…”

Section: Introductionmentioning

confidence: 99%

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Norms of Truncated Toeplitz Operators and Numerical Radii of Restricted Shifts

Gorkin

Partington

2019

Comput. Methods Funct. Theory

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“…Then our computations require us to solve zB(z) = ±1. From (10) we see that the real parts of the roots satisfy…”

Section: Degree-3 Blaschke Productsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Norms of Truncated Toeplitz Operators and Numerical Radii of Restricted Shifts

Gorkin

Partington

2019

Comput. Methods Funct. Theory

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“…What happens if we connect every third point? Examples of higher-degree cases in which the numerical range is elliptical can be found in [25,29,37]. Example 3.8.…”

Section: The Poncelet Propertymentioning

confidence: 99%

“…For Poncelet ellipses inscribed in triangles, we refer to the paper [24]. For more on a Blaschke product perspective of Poncelet's theorem, see also [25,26].…”

Section: The Poncelet Propertymentioning

confidence: 99%

Numerical range and compressions of the shift

Bickel¹,

Gorkin²

2020

Complex Analysis and Spectral Theory

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The numerical range of a bounded, linear operator on a Hilbert space is a set in C that encodes important information about the operator. In this survey paper, we first consider numerical ranges of matrices and discuss several connections with envelopes of families of curves. We then turn to the shift operator, perhaps the most important operator on the Hardy space H 2 (D), and compressions of the shift operator to model spaces, i.e. spaces of the form H 2 θH 2 where θ is inner. For these compressions of the shift operator, we provide a survey of results on the connection between their numerical ranges and the numerical ranges of their unitary dilations. We also discuss related results for compressed shift operators on the bidisk associated to rational inner functions and conclude the paper with a brief discussion of the Crouzeix conjecture. Primary 47A12; Secondary 47A13, 30C15 Date: October 30, 2018.

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“…The second set of results concern Blaschke products. Starting in 2002, Daepp, Gorkin and collaborators wrote a series of papers [14,15,17,36,37,38] considering finite Blaschke products 1 of the form (for {z j } n j=1 ⊂ D := {z ∈ C : |z| < 1}, maybe not be all distinct)…”

Section: Introductionmentioning

confidence: 99%