Fixed curves near fixed points

It is shown that rational functions continuous on the unit circle and of maximal equios-cillation order are either nonconstant finite Blaschke products or nonconstant inverted finite Blaschke products.

It is shown that a finite Blaschke product with finite poles, has a nonzero residue. The proofs for the two types of Blaschke products are essentially different.

The new framework of random polynomials developed by R. Pemantle, I. Rivin and the late O. Schramm has been studied in this thesis. The strong Pemantle-Rivin conjecture asks whether for random polynomials with independent and identically distributed zeros with a common probability distribution μon the complex plane, the empirical measures of their critical points would converge weakly to μ almost surely. This convergence question has connection with geometry of polynomials. S. D. Subramanian confirmed the conjecture whenμis a non-uniform distribution supported in the unit circle ∂D.
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\nIn this thesis, the conjecture has been extended to considering not only the critical points of the random polynomials, but also the zeros of their higher order, polar and Sz.-Nagy's generalized derivatives. The case thatμ(uniform or not) is supported in ∂D has been studied, where the derivative of each order has been proved to satisfy the conjecture. Subramanian's work has thereby been completed plus generalization. The same almost sure weak convergence has also been shown for polar and Sz.-Nagy's generalized derivatives, under some mild conditions. In particular, the result on polar derivative is the crux of filling up Subramanian's missing case of uniform μ.
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\nMeanwhile, the original Pemantle{Rivin conjecture asks about convergence in probability instead of the aforesaid stronger almost sure convergence. Z. Kabluchko fully solved this conjecture. In this thesis, his methodologies have been adapted to prove the analogous conjecture for random finite Blaschke products.
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\nMore precisely, for random finite Blaschke products with independent and identically distributed zeros with a common probability distributionμon the unit disc, the empirical measures of their critical points have been shown to converge weakly toμin probability. Consequently, this work contributes the very first probabilistic result to T. W. Ng and C. Y. Tsang's polynomial-finite-Blaschke-product dictionary.
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\nWhile the above works address probabilistic problems about zero distribution of polynomials and finite Blaschke products (regarded as non-euclidean polynomials by J. L. Walsh), the other part of this thesis is an application of the BKK theory in deterministic polynomial systems.
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\nThe numbers of various configurations for vortex dynamics in an ideal plane fluid without background flow had been investigated by many researchers for decades. In this thesis, fixed equilibria have been studied in the presence of a steady, incompressible and irrotational background flow. A more physically significant definition of a fixed equilibrium configuration has been suggested. Under this new definition, an attainable generic upper bound for their number has been found (in terms of the degree of the non-constant polynomial that determines the background flow and of the sizes of vortex species).
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\nOur result is established by transforming the rational function system arisen from fixed equilibria into a polynomial system, whose form is good enough to apply the aforesaid BKK theory (named after D. N. Bernshtein, A. G. Khovanskii and A. G. Kushnirenko) to show the finiteness of its number of solutions. Having this finiteness, the required bound follows from Bézout's theorem.

Consider the multiplication operator MB in $L^2(\mathbb{T})$, where the symbol B is a finite Blaschke product. In this article, we characterize the commutant of MB in $L^2(\mathbb{T})$. As an application of this characterization result, we explicitly determine the class of conjugations commuting with $M_{z^2}$ or making $M_{z^2}$ complex symmetric by introducing a new class of conjugations in $L^2(\mathbb{T})$. Moreover, we analyse their properties while keeping the whole Hardy space, model space and Beurling-type subspaces invariant. Furthermore, we extended our study concerning conjugations in the case of finite Blaschke products.

An explicit construction using Riemann surfaces and Brownian motion is given for an inner function in the unit disc which is not a finite Blaschke product yet belongs to the little Bloch space ${\mathcal {B}_0}$. In addition to showing how an inner function can meet the geometric conditions for ${\mathcal {B}_0}$, this example settles an open question concerning the finite ranges of inner functions: the values which it takes only finitely often are dense in the disc.

This paper gives a note on weighted composition operators on the weighted Bergman space, which shows that for a fixed composition symbol, the weighted composition operators are bounded on the weighted Bergman space only with bounded weighted symbols if and only if the composition symbol is a finite Blaschke product.

In this thesis, we characterize the Jacobson radical of the semicrossed product A(D)×α Z+ of the disk algebra by an endomorphism α where α is defined by the composition with a finite Blaschke product φ. Precisely, the Jacobson radical is the set of those whose 0th Fourier coefficient is identically zero and whose kth Fourier coefficient vanishes on the set of recurrent points of φ. Moreover, if φ is elliptic, i.e., φ has a fixed point in the open unit disc, then the Jacobson radical coincides with the set of quasinilpotent elements.

Uniform algebras that can be expressed as inductive limits of standard simpler algebras are of particular interest. For instance, some G-disc algebras are inductive limits of sequences of disc algebras, connected with finite Blaschke products, called also Blaschke inductive limit algebras. Here we show, among others, that only G-disc algebras, and the spaces \( H^\infty (\mathbb{D}_G ) \) with G ⊂ ℚ, can be expressed as limits of countable inductive sequences of algebras of type \( A(\mathbb{D}) \) and H∞ correspondingly. We study also inductive limits of sequences of spaces of type H∞ and prove corresponding corona theorems. Further, we establish relationships between Bourgain algebras of coordinate algebras, and the Bourgain algebra of their inductive limit, and also between H∞-spaces on \( \mathbb{D} \), as coordinate algebras in an inductive sequence, and their inductive limit. While we state all results for general shift-invariant algebras AS, they apply automatically to the particular cases of algebras APS of almost periodic functions, and of H S ∞ -algebras.

The classical Gauss—Lucas theorem describes the location of the critical points of a polynomial. There is also a hyperbolic version, due to Walsh, in which the role of polynomials is played by finite Blaschke products on the unit disk. We consider similar phenomena for generic inner functions, as well as for certain “locally inner” self-maps of the disk. More precisely, we look at a unit-norm function f ∈ H∞ that has an angular derivative on a set of positive measure (on the boundary) and we assume that its inner factor, I, is nontrivial. Under certain conditions to be discussed, it follows that f′ must also have a nontrivial inner factor, say J, and we study the relationship between the boundary singularities of I and J. Examples are furnished to show that our sufficient conditions cannot be substantially relaxed.

The little Bloch space, B0, is the space of all holomorphic functions f on the unit disk such that limlzl?1 lf'(z)l (1- lzl2) = 0. Finite Blaschke products are clearly in B0, but examples of infinite products in B0 are more difficult to obtain (there are now several constructions due to Sarason, Stephenson and the author, among others). Stephenson has asked whether B0 contains an infinite, indestructible Blaschke product, i.e., a Blaschke product B so that (B(z) - a)/(1 - âB(z)), is also a Blaschke product for every element a I D. In this paper we give an afirmative answer to his question by constructing such a Blaschke product. We also answer a question of Carmona and Cufi by constructing a VMO function, f, so that ll f ll8 = 1 and whose range set, R(f,a) = {w : there exists zn ? a, f(zn) = w}, equals the open unit disk for every a I T.

In this note we give an elementary proof of a theorem first proved by J. A. Erdos [3]. This theorem, which is the main result of [3], states that every noninvertible n × n matrix is a finite product of matrices M with the property that M = M . (These are known as idempotent matrices. Noninvertible matrices are also called singular matrices.) An alternative formulation of this result reads: every noninvertible linear mapping of a finite dimensional vector space is a finite product of idempotent linear mappings α, linear mappings that satisfy α = α. This result was motivated by a result of J. M. Howie asserting that each selfmapping α of a nonempty finite set X with image size at most |X| − 1 (which occurs precisely when α is noninvertible) is a product of idempotent mappings. We shall see that Erdos’s theorem is a consequence of Howie’s result. Together the papers [3] and [4] are cited in over one hundred articles, dealing with subjects including universal algebra, ring theory, topology, and combinatorics. Since its publication, various proofs of the result in [3] have appeared. For example, a semigroup theoretic proof appears in [1, pp. 121-131] and linear operator theory is used to prove the theorem in [2]. Here we give a new proof using a basic combinatorial argument. Unlike the previous proofs our argument involves only elementary results from linear algebra and one basic result concerning permutations. On the way to proving the main result of this note we provide a short proof of Howie’s result. Throughout this paper X signifies an arbitrary nonempty finite set. If α : A → X, where A is a subset of X, then A is the domain of α; we denote this set by dom(α). Naturally, the set α(A) is called the image of α and is denoted by im(α). Recall that a mapping α is injective (or one-to-one) if α(x) 6= α(y) for all x and y in dom(α) with x 6= y. Let TX denote the set of all mappings from X to X with domain X. We note that this set is closed under composition of mappings and that this composition is associative. We now define one of the most important notions we require in the proofs in this note. For a mapping α : dom(α) → X we say that α is a restriction of an element β of TX if β and α agree on the domain of α. In other words, β(x) = α(x) for all x in dom(α). For x and y in X we denote the transposition that fixes every point of X other than x or y and that maps x to y, and vice versa, by (x y).

A Solution of max |<i>a</i>₁| Using Finite Blaschke Products

Let F and G be finite Blaschke products of degree greater than one. Let p be the Wolff point of F such that either or . We prove that F commutes with G if and only if for some positive integers i,j. Also, if F and G have the same degree and commute, we prove that either F = G or G = ωF, where ω is a root of the unity.

In this paper we consider the following ‘Toeplitz completion’ problem: Complete the unspecified analytic Toeplitz entries of the partial block Toeplitz matrix $ \begin{linenomath} A:=\begin{bmatrix} T_{\overline\psi_1}&amp; ?\\[4pt] \T_{\overline\psi_2} \end{bmatrix} \end{linenomath} $ to make A hyponormal, where ψi∈H∞ is a non-constant rational function for i=1,2.

For any finite Blaschke product B, there is an injective analytic map $$\varphi :{\mathbb {D}}\rightarrow {\mathbb {C}}$$ and a polynomial p of the same degree as B such that $$B=p\circ \varphi $$ on $${\mathbb {D}}$$ . Several proofs of this result have been given over the past several years, using fundamentally different methods. However, even for low-degree Blaschke products, no method has hitherto been developed to explicitly compute the polynomial p or the associated conformal map $$\varphi $$ . In this paper, we show how these functions may be computed for a Blaschke product of degree at most three, as well as for Blaschke products of arbitrary finite degree whose zeros are equally spaced on a circle centered at the origin.