Construction of an Inner Function in the Little Bloch Space

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.

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.

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.

Let ϕ and ψ be holomorphic self-maps of the unit disk, and denote by C ϕ , C ψ the induced composition operators. This paper gives some simple estimates of the essential norm for the difference of composition operators C ϕ −C ψ from Bloch spaces to Bloch spaces in the unit disk. Compactness of the difference is also characterized.

We call a sequence {zn} in the open unit disk Δalmost thin if \({\mathop {\lim \sup }\limits_{n \to \infty } }\Pi {\left( {z_{n} } \right)} = 1,\) where Π(zn) is the product of the pseudo-hyperbolic distances between zn and the other elements in the sequence. We call the sequence thick if \({\mathop {\lim }\limits_{n \to \infty } }\Pi {\left( {z_{n} } \right)} = 0.\) In this paper, we use elementary complex variables methods to prove a sharper form of a characterization of the symbols of the isometric composition operators on the Bloch space proved by Martin and Vukotic in terms of the preimages of one-point sets. Specifically, we show that a bounded analytic function φ from Δ into itself has Bloch semi-norm equal to one (which is the largest possible) if and only if it is a conformal automorphism of Δ or if the preimage of each point of Δ contains a sequence along which the hyperbolic derivative of φ approaches one (condition (M’)). In particular, the preimage of each point of Δ is an almostthin sequence. Using similar techniques we also show that a bounded analytic function f is in the little Bloch space if and only if for each \(a\in \mathbb{C}\) such that \(f^{-1}(a)=\{z_{n}:n\in \mathbb{N}\}\) is infinite $$ {\mathop {\lim }\limits_{n \to \infty } }\left(1 - |z_{n} |^{2} \right)|f'{\left( {z_{n} } \right)}| = 0. $$ Consequently, an infinite Blaschke product whose left composition by a conformal automorphism of Δ is always a Blaschke product belongs to the little Bloch space if and only if the preimage of every point in Δ is a thick sequence. We give an example to show that this characterization does not extend to unbounded Bloch functions. We also prove an analogous characterization of the little normal functions. Finally, we give a sufficient condition on a holomorphic self-map of the polydisk Δn to be the symbol of an isometric composition operator on the Bloch space of Δn and discuss a higher-dimensional analogue of condition (M’).

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.

We consider a class of spaces of analytic functions on the unit disc which are Möbius invariant and whose topology is essentially determined by a conformal invariant seminorm. Standard examples of such spaces are the Bloch space, BMOA, the Dirichlet spaces and their recent generalizations QK , which make the object of our interest. We prove a general inequality for the seminorms of dilated functions, radial growth estimates, embedding theorems in Lp -spaces on the unit disc, as well as integral estimates of exponentials of functions in such spaces. Finally, we discuss some properties of the inner-outer factorization for those QK spaces which are contained in the Nevanlinna class.

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.

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.

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.

On the Bergman space in the unit disk, we study the intertwining relation for Volterra type operators, whose intertwining operator is a composition operator. We also investigate the “compact” intertwining relations for Volterra type operators. As obvious consequences, the essential commutativity of Volterra type and composition operators are characterized. At the end of the paper, we find a new connection between the Bergman space and little Bloch space through this essential commutativity.

Let {mathbb {D}} be the unit disc in {mathbb {C}}. If mu is a finite positive Borel measure on the interval [0, 1) and f is an analytic function in {mathbb {D}}, f(z)=sum _{n=0}^infty a_nz^n (zin {mathbb {D}}), we define Cμ(f)(z)=∑n=0∞μn∑k=0nakzn,z∈D,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\mathcal {C}}_\\mu (f)(z)= \\sum _{n=0}^\\infty \\mu _n\\left( \\sum _{k=0}^na_k\\right) z^n,\\quad z\\in {\\mathbb {D}}, \\end{aligned}$$\\end{document}where, for nge 0, mu _n denotes the n-th moment of the measure mu , that is, mu _n=int _{[0, 1)}t^ndmu (t). In this way, {mathcal {C}}_mu becomes a linear operator defined on the space {mathrm{Hol}}({mathbb {D}}) of all analytic functions in {mathbb {D}}. We study the action of the operators {mathcal {C}}_mu on distinct spaces of analytic functions in {mathbb {D}}, such as the Hardy spaces H^p, the weighted Bergman spaces A^p_alpha , BMOA, and the Bloch space {mathcal {B}}.

Let be the unit disk in the complex plane. We define to be the little Bloch space of functions f analytic in which satisfy lim|z|→1(1 - |z|2)|f'(z)| = 0. If is analytic then the composition operator C φ : f ↦ f ∘ φ is a continuous operator that maps into itself. In this paper, we show that the compactness of C φ , as an operator on , can be modelled geometrically by its principal eigenfunction. In particular, under certain necessary conditions, we relate the compactness of C φ to the geometry of , where σ satisfies Schöder's functional equation σ ∘ φ = φ'(0)σ.2000 Mathematics Subject Classification: Primary 30D05; 47B33 Secondary 30D45.

In this paper, we characterise the analytic functions ϕ mapping the open unit disk ▵ into itself whose induced composition operator Cϕ: f ↦ f ∘ ϕ is an isometry on the Bloch space. We show that such functions are either rotations of the identity function or have a factorisation ϕ = gB where g is a non-vanishing analytic function from Δ into the closure of ▵, and B is an infinite Blaschke product whose zeros form a sequence{zn} containing 0 and a subsequence satisfying the conditions , and

Aleksandrov, Anderson and Nicolau have found examples of inner functions that are in the little Bloch space with a specific rate of convergence to zero. As a corollary they obtain positive singular measures defined in the boundary of the unit disc that are simoultaneously symmetric and Kahane. Nevertheless their construction is very indirect. We give an explicit example of such measures by means of a martingale argument.