Ellipses and compositions of finite Blaschke products

Gorkin, Pamela; Wagner, Nathan A.

doi:10.1016/j.jmaa.2016.01.067

Cited by 18 publications

(13 citation statements)

References 11 publications

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“…For a Blaschke product B of degree 4, the interior curve I B is an ellipse if and only if B is a composition of two Blaschke products of degree 2. See [Fuj17], and also see [GW17] for the relationship between this result and the numerical range of shift perators.…”

Section: Examplesmentioning

confidence: 88%

Interior and exterior curves of finite Blaschke products

Fujimura

2018

Journal of Mathematical Analysis and Applications

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For a Blaschke product B of degree d and λ on ∂D, let λ be the set of lines joining each distinct two preimages in B −1 (λ). The envelope of the family of lines { λ } λ∈∂D is called the interior curve associated with B. In 2002, Daepp, Gorkin, and Mortini proved the interior curve associated with a Blaschke product of degree 3 forms an ellipse. While let L λ be the set of lines tangent to ∂D at the d preimages B −1 (λ) and the trace of the intersection points of each two elements in L λ as λ ranges over the unit circle is called the exterior curve associated with B. In 2017, the author proved the exterior curve associated with a Blaschke product of degree 3 forms a non-degenerate conic.In this paper, for a Blaschke product of degree d, we give some geometrical properties that lie between the interior curve and the exterior curve.

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Section: Examplesmentioning

confidence: 88%

Interior and exterior curves of finite Blaschke products

Fujimura

2018

Journal of Mathematical Analysis and Applications

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“…The model space K B of B is the n-dimensional linear space of all rational functions p/q with denominator q(z) := (1 − λ k z) and deg p ≤ n − 1 (see [7,Chapter 12] or [11], for instance).…”

Section: Blaschke Kippenhahn and Ponceletmentioning

confidence: 99%

On partial isometries with circular numerical range

Wegert¹,

Spitkovsky²

2021

Preprint

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In their LAMA'2016 paper Gau, Wang and Wu conjectured that a partial isometry A acting on C n cannot have a circular numerical range with a non-zero center, and proved this conjecture for n ≤ 4. We prove it for operators with rank A = n − 1 and any n.The proof is based on the unitary similarity of A to a compressed shift operator S B generated by a finite Blaschke product B. We then use the description of the numerical range of S B as intersection of Poncelet polygons, a special representation of Blaschke products related to boundary interpolation, and an explicit formula for the barycenter of the vertices of Poncelet polygons involving elliptic functions.

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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%

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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Ellipses and compositions of finite Blaschke products

Cited by 18 publications

References 11 publications

Interior and exterior curves of finite Blaschke products

Interior and exterior curves of finite Blaschke products

On partial isometries with circular numerical range

Numerical range and compressions of the shift

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