Four Theorems with Their Foci on Ellipses

Abstract

We consider several seemingly unrelated theorems, each with the word ellipse in the statement: Siebeck’s theorem, the elliptical range theorem, Poncelet’s theorem, and the Blaschke ellipse theorem. Though Siebeck’s theorem is a geometric statement about complex functions, we use linear algebra and the numerical range of a matrix to provide a proof of the theorem. Poncelet’s theorem, from projective geometry, and rational functions known as Blaschke products provide some surprising additional connections.

MSC:

• Primary 15A60, Secondary 30J10

Acknowledgment

The author is grateful to the referees and the editor for suggestions that have improved the manuscript. In addition, the author would like to thank Thomas Cassidy, Ruth Cassidy, Ulrich Daepp, and Karl Voss for their valuable comments.

The author’s research was supported in part by Simons Foundation Grant no. 243653.

Notes

1 See su.se/polopoly_fs/1.229312.1426783194!/menu/standard/file/marden.pdf, accessed May 3, 2017.

2 We consider only matrices, but this result is true for operators on a Hilbert space.

3 See dankalman.net/AUhome/pdffiles/mardenMH.pdf, accessed May 4, 2017.

4 For the other sides, we would choose x₁ in the kernel of $P^{\star }{E}_{1}P$ or x₂ in the kernel of $P^{\star }{E}_{2}P$.

5 This proof works in general and constructs a Blaschke product identifying points $z_1,\dots ,z_n\in \mathbb{T}$.

6 This is the subject of a forthcoming paper by K. Bickel, T. Tran, and the author.

7 See mathforum.org/mathimages/index.php/Envelope.

Additional information

Notes on contributors

Pamela Gorkin

Pamela Gorkin is a professor of mathematics at Bucknell University. She studied at Michigan State University under Sheldon Axler, who studied under Don Sarason, who studied under Paul Halmos, who studied the numerical range. Her research as well as her teaching have been heavily influenced by these three great mathematicians and teachers.