Title

Moebius transformations and Blaschke products: the geometric connection.

Publication Date

2017

Journal

Linear Algebra Appl.

Volume

516

First Page

186

Last Page

211

Department

Mathematics

Abstract

Let B be a degree-n Blaschke product and, for a complex number l of modulus 1, let z1l, ... znl ordered according to increasing argument, denote the (distinct) solutions to B(z) - l = 0. Then there is a smooth curve C such that for each l the line segments joining zjl and z(j+1)l are tangent to C. We study the situation in which C is an ellipse and describe the relation to the action of the points zjl under elliptic disk automorphisms. These results provide a condition for the numerical range of a compressed shift operator with finite Blaschke symbol to be an elliptical disk. We also consider infinite Blaschke products and the action of parabolic and hyperbolic disk automorphisms

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