Rational tetra-inner functions and the special variety of the tetrablock

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:506

Article

Alsalhi, Omar M. O.¹  Lykova, Zinaida A.² 

[1] Umm Alqura Univ, Alleith Univ Coll, Al Leith, Saudi Arabia

[2] Newcastle Univ, Sch Math Stat & Phys, Newcastle Upon Tyne NE1 7RU, Tyne & Wear, England

关键词: Inner functions;    Tetrablock;    Convexity;    Extreme point;    Distinguished boundary;   

DOI  :  10.1016/j.jmaa.2021.125534

来源: Elsevier

【 摘 要 】

The set (E) over bar = {x is an element of C-3 : 1 - x(1)z - x(2)w + x(3)zw not equal 0 whenever vertical bar z vertical bar < 1, vertical bar w vertical bar < 1} is called the tetrablock and has intriguing complex-geometric properties. It is polynomially convex, nonconvex and starlike about 0. It has a group of automorphisms parametrised by Aut D x Aut D x Z(2) and its distinguished boundary b (E) over bar is homeomorphic to the solid torus (D) over bar x T. It has a special subvariety R-(E) over bar = {(x(1), x(2), x(3)) is an element of(E) over bar : x(1)x(2) = x(3)}, called the royal variety of (E) over bar. R-(E) over bar is a complex geodesic of E and it is invariant under all automorphisms of (E) over bar. We make use of these geometric properties of (E) over bar to develop an explicit structure theory for the rational maps from the unit disc D to (E) over bar that map the unit circle T to the distinguished boundary b (E) over bar of (E) over bar. Such maps are called rational (E) over bar -inner functions. We call the points lambda is an element of D such that x(lambda) is an element of R-(E) over bar the royal nodes of x. We describe the construction of rational (E) over bar -inner functions of prescribed degree from the zeros of x(1) and x(2) and the royal nodes of x. The proof of this theorem is constructive: it gives an algorithm for the construction of a 3-parameter family of such functions x subject to the computation of Fejer-Riesz factorizations of certain non-negative functions on the circle. We show that, for each nonconstant rational E-inner function x, either x((D) over bar) subset of R-(E) over bar boolean AND (E) over bar or x((D) over bar) meets R-(E) over bar exactly deg(x) times. We study convex subsets of the set J of all rational (E) over bar -inner functions and extreme points of J. We show that whether a rational (E) over bar -inner function x is an extreme point of J depends on how many royal nodes of x lie on T. (C) 2021 The Author(s). Published by Elsevier Inc.

【 授权许可】

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