Möbius transformations and Blaschke products: The geometric connection

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• 作者：Ulrich Daepp^a ; ^{udaepp@bucknell.edu} ; Pamela Gorkin^a ; ¹ ; ^{pgorkin@bucknell.edu} ; Andrew Shaffer^b ; ^{ashaffer@bucknell.edu} ; Karl Voss^a ; ^{kvoss@bucknell.edu}
• 关键词：30J10 ; 30D05
• 刊名：Linear Algebra and its Applications
• 出版年：2017
• 出版时间：1 March 2017
• 年：2017
• 卷：516
• 期：Complete
• 页码：186-211
• 全文大小：967 K
• 卷排序：516

文摘

Let B be a degree-n   Blaschke product and, for λ∈Tλ∈T, let z1,λ,…,zn,λz1,λ,…,zn,λ, ordered according to increasing argument, denote the (distinct) solutions to B(z)−λ=0B(z)−λ=0. Then there is a smooth curve CC such that for each λ   the line segments joining zj,λzj,λ and zj+1,λzj+1,λ are tangent to CC. We study the situation in which CC is an ellipse and describe the relation to the action of the points zj,λzj,λ 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.