Complex Unit Disk Research Articles

Möbius’s functional equation and Schur’s lemma with applications to the complex unit disk

Mobius addition is defined on the complex open unit disk by $$\begin{aligned} a\oplus _M b = \dfrac{a+b}{1+\bar{a}b} \end{aligned}$$ and Mobius’s exponential equation takes the form $$L(a\oplus _M b) = L(a)L(b)$$ , where L is a complex-valued function defined on the complex unit disk. In the present article, we indicate how Mobius’s exponential equation is connected to Cauchy’s exponential equation. Mobius’s exponential equation arises when one determines the irreducible linear representations of the unit disk equipped with Mobius addition, considered as a nonassociative group-like structure. This suggests studying Schur’s lemma in a more general setting.

A sharp bound on the Lebesgue constant for Leja points in the unit disk

We give a sharp bound for the Lebesgue constant associated with Leja sequences in the complex unit disk, confirming a conjecture made by Calvi and Phung in 2011.

On algebras generated by Toeplitz operators and their representations

We study Banach and C⁎-algebras generated by Toeplitz operators acting on weighted Bergman spaces Aλ2(B2) over the complex unit ball B2⊂C2. Our key point is an orthogonal decomposition of Aλ2(B2) into a countable sum of infinite dimensional spaces, each one of which can be identified with a differently weighted Bergman space Aμ2(D) over the complex unit disk D. Moreover, all elements of the above algebras leave each of the summands in the above decomposition invariant and their restriction to each level acts as a compact perturbation of a Toeplitz operator on Aμ2(D).The symbols of the generating Toeplitz operators are chosen to be suitable extensions to B2 of families S of bounded functions on D. Symbol classes S that generate important classical commutative and non-commutative Toeplitz algebras in L(Aμ2(D)) are of particular interest. In this paper we discuss various examples. In the case of S=C(D‾) and S=C(D‾)⊗L∞(0,1) we characterize all irreducible representations of the resulting Toeplitz operator C⁎-algebras. Their Calkin algebras are described and index formulas are provided.

A q-analogue of the weighted Bergman space on the disk and associated second q-Bargmann transform

We provide a q-analogue of the classical weighted Bergman space on the complex unit disk and we give the explicit expression of the corresponding reproducing kernel function. Moreover, we give a q-analogue of the second Bargmann integral transform introduced by V. Bargmann in [4, p. 203]. We show that it defines a unitary integral transform from the Hilbert space on the nonnegative real half line spanned by the q-Laguerre polynomials with respect to the weight function xαexpq⁡(−(1−q)x), onto the considered weighted q-Bergman Hilbert space.

Königs eigenfunction for composition operators on Bloch and H∞ type spaces

We discuss when the Königs eigenfunction associated with a non-automorphic selfmap of the complex unit disc that fixes the origin belongs to Banach spaces of holomorphic functions of Bloch and H∞ type. In the latter case, our characterization answers a question of P. Bourdon. Some spectral properties of composition operators on H∞ for unbounded Königs eigenfunction are obtained.

Composition operators between Bergman spaces of logarithmic weights

Let [Formula: see text] be the composition operator induced by a holomorphic self-map φ of the open complex unit disk. In this paper, a necessary and sufficient condition for the boundedness of [Formula: see text] from one weighted Bergman space of logarithmic weight into another is described in terms of a growth condition of a generalized counting function for φ. We make use of a new integral representation of a modified counting function which depends on log-convexity of the weight function as well as some estimates for the norm of the weighted Bergman space.

2D-Zernike Polynomials and Coherent State Quantization of the Unit Disc

Using the orthonormality of the 2D-Zernike polynomials, reproducing kernels, reproducing kernel Hilbert spaces, and ensuring coherent states attained. With the aid of the so-obtained coherent states, the complex unit disc is quantized. Associated upper symbols, lower symbols and related generalized Berezin transforms also obtained. A number of necessary summation formulas for the 2D-Zernike polynomials proved.

Analytic Campanato Spaces and Their Compositions

This paper is devoted to characterizing the analytic Campanato spaces $\mathcal{AL}_{p,\eta}$ (including the analytic Morrey spaces, the analytic John-Nirenberg space, and the analytic Lipschitz/H\older spaces) on the complex unit disk $\mathbb D$ in terms of the M\obius mapping and the Littlewood-Paley form, and consequently their compositions with the analytic self-maps of $\mathbb D$.

Statistical Properties of the Excited Binomial States for the Pseudoharmonic Oscillator

In this paper we have studied the binomial states and also the excited binomial states defined in terms of the Fock-basis vectors of the pseudoharmonic oscillator. We have demonstrated that outside of the behavior at the harmonic limit, when the binomial states lead to the coherent states of the one dimensional harmonic oscillator, these states have in fact all the characteristics of the coherent states defined on the complex unit disk. We have calculated the expectation values and the Mandel parameter (as well as their thermal analogue) which give us information on their statistical behavior. All the obtained relations tend, at the harmonic limit, to the corresponding relations for the one dimensional harmonic oscillator.

De Branges–Rovnyak Realizations of Operator-Valued Schur Functions on the Complex Right Half-Plane

We give a controllable energy-preserving and an observable co-energy-preserving de Branges-Rovnyak functional model realization of an arbitrary given operator Schur function defined on the complex right-half plane. We work the theory out fully in the right-half plane, without using results for the disk case, in order to expose the technical details of continuous-time systems theory. At the end of the article, we make explicit the connection to the corresponding classical de Branges-Rovnyak realizations for Schur functions on the complex unit disk.

COMPACT INTERTWINING RELATIONS FOR COMPOSITION OPERATORS BETWEEN THE WEIGHTED BERGMAN SPACES AND THE WEIGHTED BLOCH SPACES

Abstract. We study the compact intertwining relations for compositionoperators, whose intertwining operators are Volterra type operators fromthe weighted Bergman spaces to the weighted Bloch spaces in the unitdisk. As consequences, we find a new connection between the weightedBergman spaces and little weighted Bloch spaces through this relations. 1. IntroductionIf X and Y are two Banach spaces, the symbol B(X,Y ) denotes the col-lection of all bounded linear operators from X to Y . Let K(X,Y ) be thecollection of all compact elements of B(X,Y ), and Q(X,Y) be the quotientset B(X,Y )/K(X,Y ).For linear operators A ∈ B(X,X), B ∈ B(Y,Y ) and T ∈ B(X,Y ), thephrase “T intertwines A and B in Q(X,Y)” (or “T intertwines A and B com-pactly”) means that(1.1) TA = BT mod K(X,Y ) with T 6= 0 .Notation A ∝ K B (T) represents the relation in equation (1.1). In fact, if T isan invertible operator on X, then the relation ∝ K is symmetric.We denote the class of all holomorphic functions on the complex unit disk Dby H(D), and the collection of all the holomorphic self mappings of Dby S(D).Let α > −1, 0 0. These settingsof α,β,γ and p are valid in the following context unless specifications.

On the Lebesgue constant of Leja sequences for the complex unit disk and of their real projection

We consider Leja sequences of points for polynomial interpolation on the complex unit disk U and the corresponding sequences for polynomial interpolation on the real interval [−1,1] obtained by projection. It was proved by Calvi and Phung in Calvi and Phung (2011, 2012) [3,4] that the Lebesgue constants for such sequences are asymptotically bounded in O(klogk) and O(k3logk) respectively, where k is the number of points. In this paper, we establish the sharper bound 5k2logk in the real interval case. We also give sharper bounds in the complex unit disk case, in particular 2k. Our motivation for producing such sharper bounds is the use of these sequences in the framework of adaptive sparse polynomial interpolation in high dimension.

Multiplicative mimicry and improvements to the Pólya–Vinogradov inequality

We study exponential sums whose coefficients are completely multiplicative and belong to the complex unit disc. Our main result shows that such a sum has substantial cancellation unless the coefficient function is essentially a Dirichlet character. As an application we improve current bounds on odd-order character sums. Furthermore, conditionally on the generalized Riemann hypothesis we obtain a bound for odd-order character sums which is best possible.

Berezin transform and Weyl-type unitary operators on the Bergman space

For D the open complex unit disc with normalized area measure, we consider the Bergman space La(D) of square-integrable holomorphic functions on D. Induced by the group Aut(D) of biholomorphic automorphisms of D, there is a standard family of Weyl-type unitary operators on La(D). For all bounded operators X on La(D), the Berezin transform X is a smooth, bounded function on D. The range of the mapping Ber: X → X is invariant under Aut(D). The “mixing properties” of the elements of Aut(D) are visible in the Berezin transforms of the induced unitary operators. Computations involving these operators show that there is no real number M > 0 with M‖ X‖∞ ≥ ‖X‖ for all bounded operators X and are used to check other possible properties of X. Extensions to other domains are discussed.

The Schwarz-Pick lemma for slice regular functions

The celebrated Schwarz-Pick lemma for the complex unit disk is the basis for the study of hyperbolic geometry in one and in several complex variables. In the present paper, we turn our attention to the quaternionic unit ball B. We prove a version of the Schwarz-Pick lemma for self-maps of B that are slice regular, according to the definition of Gentili and Struppa. The lemma has interesting applications in the fixed-point case, and it generalizes to the case of vanishing higher order derivatives.

The Lindelöf principle for holomorphic functions of infinitely many variables

Let D be a convex bounded domain in a complex Banach space. A holomorphic function f : D → ℂ is called a normal function if the family ℱ f = {f ○ ϕ : ϕ ∈ 𝒪(Δ, D)} forms a normal family in the sense of Montel (here 𝒪(Δ, D) denotes the set of all holomorphic maps from the complex unit disc into D). Let {x n } be a sequence of points in D which tends to a boundary point ξ ∈ ∂D, such that lim n→∞ f(x n ) = l for some The sufficient conditions on a sequence {x n } of points in D and a normal holomorphic function f are given for f to have the admissible limit value l, thus extending the result obtained by Bagemihl and Seidel. This result is used to draw a new Lindelöf principle in the holomorphic function theory of infinite many variables. The results in this article are improvements of earlier results of Cirka, Dovbush, Cima and Krantz.

On the construction of uniformly convergent disk polynomial expansions

We present a method to derive explicit uniformly convergent disk polynomial expansions for real-analytic functions on the complex unit disk. As an application, we deduce the spherical harmonic expansion of the Poisson–Szego kernel for the unit ball in \({\mathbb{C}^q}\) .

Asymptotics for polynomials orthogonal over the unit disk with respect to a positive polynomial weight

We derive asymptotics for polynomials orthogonal over the complex unit disk with respect to a weight of the form | h ( z ) | 2 , with h ( z ) a polynomial without zeros in | z | < 1 . The behavior of the polynomials is established at every point of the complex plane. The proofs are based on adapting to the unit disk a technique of J. Szabados for the asymptotic analysis of polynomials orthogonal over the unit circle with respect to the same type of weight.

On normal and non-normal holomorphic functions on complex Banach manifolds

Let X be a complex Banach manifold. A holomorphic function f : X → C is called a normal function if the family F f = { f ◦φ : φ ∈ O( , X)} forms a normal family in the sense of Montel (here O( , X) denotes the set of all holomorphic maps from the complex unit disc into X ). Characterizations of normal functions are presented. A sufficient condition for the sum of a normal function and non-normal function to be non-normal is given. Criteria for a holomorphic function to be non-normal are obtained. These results are used to draw one interesting conclusion on the boundary behavior of normal holomorphic functions in a convex bounded domain D in a complex Banach space V . Let {xn} be a sequence of points in D which tends to a boundary point ξ ∈ ∂ D such that limn→∞ f (xn) = L for some L ∈ C. Sufficient conditions on a sequence {xn} of points in D and a normal holomorphic function f are given for f to have the admissible limit value L , thus extending the result obtained by Bagemihl and Seidel. Mathematics Subject Classification (2000): 32A18 (primary).

On Hyperbolic Divided Differences and the Nevanlinna-Pick Problem

Starting from the notion of the complex pseudo-hyperbolic distance and the hyperbolic difference quotient introduced by A. F. Beardon and D. Minda in [1], we define hyperbolic divided differences for unimodularly bounded holomorphic functions in the complex unit disk and investigate their mapping properties. In particular, we show that they operate on Blaschke products in the same way as the ordinary divided differences act on polynomials. As a simple corollary we obtain a multi-point Schwarz-Pick Lemma.