Ellipses and Finite Blaschke Products

Daepp, Ulrich; Gorkin, Pamela; Mortini, Raymond

doi:10.2307/3072367

Cited by 26 publications

(31 citation statements)

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“…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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“…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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“…One algorithm for constructing D is presented in [12]. Up to rotation, there is only one such Blaschke product D; see [7,Theorem 2]. It then remains to check the derivative condition.…”

Section: Examplementioning

confidence: 99%

Clark measures and a theorem of Ritt

Chalendar¹,

Gorkin²,

Partington³

et al. 2018

Math. Scand.

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We determine when a finite Blaschke product $B$ can be written, in a non-trivial way, as a composition of two finite Blaschke products (Ritt's problem) in terms of the Clark measure for $B$. Our tools involve the numerical range of compressed shift operators and the geometry of certain polygons circumscribing the numerical range of the relevant operator. As a consequence of our results, we can determine, in terms of Clark measures, when two finite Blaschke products commute.

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“…In the case n = 3 the curve is an ellipse and the foci are the geometric foci of the ellipse ( [2]). …”

Section: Proposition 31 the Curve η Which Is The Envelope Of The LImentioning

confidence: 99%

Untitled

Singer

2006

Conform. Geom. Dyn.

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Abstract. A theorem of Bôcher and Grace states that the critical points of a cubic polynomial are the foci of an ellipse tangent to the sides of the triangle joining the zeros. A more general result of Siebert and others states that the critical points of a polynomial of degree N are the algebraic foci of a curve of class N − 1 which is tangent to the lines joining pairs of zeroes. We prove the analogous results for hyperbolic polynomials, that is, for Blaschke products with N roots in the unit disc.

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Ellipses and Finite Blaschke Products

Cited by 26 publications

References 0 publications

Numerical range and compressions of the shift

Numerical range and compressions of the shift

Clark measures and a theorem of Ritt

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