Complex Unit Disk Research Articles

On Möbius gyrogroup and Möbius gyrovector space

Gyrogroups are new algebraic structures that appeared in 1988 in the study of Einstein’s velocity addition in the special relativity theory. These new algebraic structures were studied intensively by Abraham Ungar. The first gyrogroup that was considered into account is the unit ball of Euclidean space \mathbb{R}^3ℝ3 endowed with Einstein’s velocity addition. The second geometric example of a gyrogroup is the complex unit disk \mathbb{D}𝔻={z ∈ \mathbb{C}: |z|<1ℂ:|z|<1}. To construct a gyrogroup structure on \mathbb{D}𝔻, we choose two elements z_1, z_2 ∈\mathbb{D}z1,z2∈𝔻 and define the Möbius addition by z_1\oplus z_2 = \frac{z_1+z_2}{1+\bar{z_1}z_2}z1⊕z2=z1+z21+z1‾z2. Then (\mathbb{D},\oplus)(𝔻,⊕) is a gyrocommutative gyrogroup. If we define r \odot xr⊙x==\frac{(1+|x|)^r - (1-|x|)^r}{(1+|x|)^r + (1-|x|)^r}\frac{x}{|x|}(1+|x|)r−(1−|x|)r(1+|x|)r+(1−|x|)rx|x|, where x ∈ \mathbb{D}x∈𝔻 and r ∈ \mathbb{R}r∈ℝ, then (\mathbb{D},\oplus,\odot)(𝔻,⊕,⊙) will be a real gyrovector space. This paper aims to survey the main properties of these Möbius gyrogroup and Möbius gyrovector space.

Noncommutative rational Clark measures

Abstract We characterize the noncommutative Aleksandrov–Clark measures and the minimal realization formulas of contractive and, in particular, isometric noncommutative rational multipliers of the Fock space. Here, the full Fock space over $\mathbb {C} ^d$ is defined as the Hilbert space of square-summable power series in several noncommuting (NC) formal variables, and we interpret this space as the noncommutative and multivariable analogue of the Hardy space of square-summable Taylor series in the complex unit disk. We further obtain analogues of several classical results in Aleksandrov–Clark measure theory for noncommutative and contractive rational multipliers. Noncommutative measures are defined as positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz algebra, the unital $C^*$ -algebra generated by the left creation operators on the full Fock space. Our results demonstrate that there is a fundamental relationship between NC Hardy space theory, representation theory of the Cuntz–Toeplitz and Cuntz algebras, and the emerging field of noncommutative rational functions.

Fatou's theorem for non-commutative measures

A classical theorem of Fatou asserts that the Radon–Nikodym derivative of any finite and positive Borel measure, μ, with respect to Lebesgue measure on the complex unit circle, is recovered as the non-tangential limits of its Poisson transform in the complex unit disk. This positive harmonic Poisson transform is the real part of an analytic function whose Taylor coefficients are in fixed proportion to the conjugate moments of μ.Replacing Taylor series in one variable by power series in several non-commuting variables, we show that Fatou's Theorem and related results have natural extensions to the setting of positive harmonic functions in an open unit ball of several non-commuting matrix-variables, and a corresponding class of positive non-commutative (NC) measures. Here, an NC measure is any positive linear functional on a certain self-adjoint unital subspace of the Cuntz–Toeplitz algebra, the C⁎−algebra generated by the left creation operators on the full Fock space.

Approximation by Convolution Polyanalytic Operators in the Complex and Quaternionic Compact Unit Balls

In this paper, by using the convolution method, we obtain quantitative results in terms of various moduli of smoothness for approximation of polyanalytic functions by polyanalytic polynomials in the complex unit disc. Then, by introducing the polyanalytic Gauss–Weierstrass operators of a complex variable, we prove that they form a contraction semigroup on the space of polyanalytic functions defined on the compact unit disk. The quantitative approximation results in terms of moduli of smoothness are then extended to the case of slice p-polyanalytic functions on the quaternionic unit ball. Moreover, we show that also in the quaternionic case the Gauss–Weierstrass operators of a quaternionic variable form a contraction semigroup on the space of polyanalytic functions defined on the compact unit ball.

Zeros of Gaussian power series, Hardy spaces and determinantal point processes

Given a sequence \((\xi _n)\) of standard i.i.d complex Gaussian random variables, Peres and Virág (in the paper “Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process” Acta Math. (2005) 194, 1-35) discovered the striking fact that the zeros of the random power series \(f(z) = \sum _{n=1}^\infty \xi _n z^{n-1}\) in the complex unit disc \({\mathbb {D}}\) constitute a determinantal point process. The study of the zeros of the general random series f(z), where the restriction of independence is relaxed upon the random variables \((\xi _n)\) is an important open problem. This paper proves that if \((\xi _n)\) is an infinite sequence of complex Gaussian random variables, such that their covariance matrix is invertible and its inverse is a Toeplitz matrix, then the zero set of f(z) constitutes a determinantal point process with the same distribution as the case of i.i.d variables studied by Peres and Virág. The arguments are based on some interplays between Hardy spaces and reproducing kernels. Illustrative examples are constructed from classical Toeplitz matrices and the classical fractional Gaussian noise.

Another Hilbert inequality and critically separated interpolation nodes

AbstractAbstract: Estimates on the condition number of Vandermonde matrices have implications on several algorithms ranging from polynomial interpolation to sparse super resolution in fluorescence microscopy. Classically, the situation is studied for monomials on real intervals, the complex unit disk, and the complex unit circle. Except for roots of unity and well separated nodes on the unit circle, the condition number grows strongly with increasing polynomial degree. Here, we show that the condition number of the Vandermonde matrix for a particular instance of critically separated nodes on the complex unit circle grows logarithmically with the polynomial degree. The proof is based on a variant of Hilbert's inequality with remainder term.

Non-commutative rational functions in the full Fock space

A rational function belongs to the Hardy space, H 2 H^2 , of square-summable power series if and only if it is bounded in the complex unit disk. Any such rational function is necessarily analytic in a disk of radius greater than one. The inner-outer factorization of a rational function r ∈ H 2 \mathfrak {r} \in H^2 is particularly simple: The inner factor of r \mathfrak {r} is a (finite) Blaschke product and (hence) both the inner and outer factors are again rational. We extend these and other basic facts on rational functions in H 2 H^2 to the full Fock space over C d \mathbb {C} ^d , identified as the non-commutative (NC) Hardy space of square-summable power series in several NC variables. In particular, we characterize when an NC rational function belongs to the Fock space, we prove analogues of classical results for inner-outer factorizations of NC rational functions and NC polynomials, and we obtain spectral results for NC rational multipliers.

Blaschke–singular–outer factorization of free non-commutative functions

By classical results of Herglotz and F. Riesz, any bounded analytic function in the complex unit disk has a unique inner–outer factorization. Here, a bounded analytic function is called inner or outer if multiplication by this function defines an isometry or has dense range, respectively, as a linear operator on the Hardy Space, H2, of analytic functions in the complex unit disk with square-summable Taylor series. This factorization can be further refined; any inner function θ decomposes uniquely as the product of a Blaschke inner function and a singular inner function, where the Blaschke inner contains all the vanishing information of θ, and the singular inner factor has no zeroes in the unit disk.We prove an exact analogue of this factorization in the context of the full Fock space, identified as the Non-commutative Hardy Space of analytic functions defined in a certain multi-variable non-commutative open unit ball.

Characterizations of the d[formula omitted]-transitive and [formula omitted]-transitive families of composition operators on Lp-spaces

In this paper, we characterize the F-transitive and the dF-transitive families of composition operators on Lp(X,B,μ), where (X,B,μ) is a σ-finite measure space. In particular, both necessary and sufficient conditions on the inducing mappings for F-transitive and dF-transitive composition operators are presented, which includes both Theorem 1.1 and Theorem 1.2 given in [1] in a more general frame. Moreover, we provide the characterizations of F-transitive and dF-transitive semigroups induced by semiflows on Lρp(X,K). And as a result, the sufficient condition of d-transitive semigroups induced by semiflows on Lρp(Ω,K) appears to be necessary, which answers a recent question posed by Kostić in [16]. Further, we provide the characterizations on the weight function ρ for F-transitive translation semigroups on Lρp(Δ,K), which extends the recent results in [10] and [13]. Some related examples of dF-transitive composition operators and C0-semigroups that are the solution semigroups of certain Cauchy problems are provided. In particular, we present the properties of dF-transitive composition operators on Lp(T,B,λ) induced by automorphisms on the complex unit disk D and show the equivalence of d-transitivity and d-mixing property for these operators.

Cosine Similarity Measure of Complex Fuzzy Sets and Robustness of Complex Fuzzy Connectives

Complex fuzzy set (CFS), as a generalization of fuzzy set (FS), is characterized by complex-valued membership degrees. By considering the complex-valued membership degree as a vector in the complex unit disk, we introduce the cosine similarity measures between CFSs. Then, we investigate some invariance properties of the cosine similarity measure. Finally, the cosine similarity measure is applied to measure the robustness of complex fuzzy connectives and complex fuzzy inference.

A metric invariant of Möbius transformations

The complex unit disk $\mathbb{D} = \{z\in\mathbb{C}\colon z < 1\}$ is endowed with Möbius addition $\oplus_M$ defined by $$ w\oplus_M z = \dfrac{w+z}{1+\overline{w}z}. $$ We prove that the metric $d_T$ defined on $\mathbb{D}$ by $d_T(w, z) = \tan^{-1} -w\oplus_M z $ is an invariant of Möbius transformations carrying $\mathbb{D}$ onto itself. We also prove that $(\mathbb{D}, d_T)$ and $(\mathbb{D}, d_P)$, where $d_P$ denotes the Poincaré metric, have the same isometry group and then classify the isometries of $(\mathbb{D}, d_T)$.

Operators affiliated to the free shift on the Free Hardy Space

The Smirnov class for the classical Hardy space is the set of ratios of bounded analytic functions in the open complex unit disk with outer denominators. This definition extends naturally to the commutative and non-commutative multi-variable settings of the Drury-Arveson space and the full Fock space over Cd. Identifying the Fock space with the free multi-variable Hardy space of non-commutative or free holomorphic functions in a non-commutative open unit ball in several matrix-variables, we prove that any closed, densely-defined operator affiliated to the right free multiplier algebra of the full Fock space acts as right multiplication by a non-commutative function in the right free Smirnov class (and analogously, replacing ‘right’ with ‘left’).

Furstenberg Systems of Bounded Multiplicative Functions and Applications

Abstract We prove a structural result for measure preserving systems naturally associated with any finite collection of multiplicative functions that take values on the complex unit disc. We show that these systems have no irrational spectrum and their building blocks are Bernoulli systems and infinite-step nilsystems. One consequence of our structural result is that strongly aperiodic multiplicative functions satisfy the logarithmically averaged variant of the disjointness conjecture of Sarnak for a wide class of zero entropy topological dynamical systems, which includes all uniquely ergodic ones. We deduce that aperiodic multiplicative functions with values plus or minus one have super-linear block growth. Another consequence of our structural result is that products of shifts of arbitrary multiplicative functions with values on the unit disc do not correlate with any totally ergodic deterministic sequence of zero mean. Our methodology is based primarily on techniques developed in a previous article of the authors where analogous results were proved for the Möbius and the Liouville function. A new ingredient needed is a result obtained recently by Tao and Teräväinen related to the odd order cases of the Chowla conjecture.

Complex Fuzzy Power Aggregation Operators

A complex fuzzy set is an extension of the fuzzy set, of which membership grades take complex values in the complex unit disk. We present two complex fuzzy power aggregation operators including complex fuzzy weighted power (CFWP) and complex fuzzy ordered weighted power (CFOWP) operators. We then study two geometric properties which include rotational invariance and reflectional invariance for these complex fuzzy aggregation operators. We also apply the new proposed aggregation operators to decision making and illustrate an example to show the validity of the new approach.

Complex Fuzzy Geometric Aggregation Operators

A complex fuzzy set is an extension of the traditional fuzzy set, where traditional [0,1]-valued membership grade is extended to the complex unit disk. The aggregation operator plays an important role in many fields, and this paper presents several complex fuzzy geometric aggregation operators. We show that these operators possess the properties of rotational invariance and reflectional invariance. These operators are also closed on the upper-right quadrant of the complex unit disk. Based on the relationship between Pythagorean membership grades and complex numbers, these operators can be applied to the Pythagorean fuzzy environment.

LMI-Based Fixed Order Output Feedback Synthesis for Two-Dimensional Mixed Continuous-Discrete-Time Systems

This paper addresses the problem of designing stabilizing fixed order output feedback controllers for two-dimensional mixed continuous-discrete-time systems. The paper starts by addressing this problem in a basic formulation, where the plant is supposed strictly proper, and the output feedback controller is supposed static. It is shown that a necessary and sufficient condition for establishing the existence of a stabilizing feedback controller can be obtained by solving two convex optimization problems with linear matrix inequality (LMI) constraints. The condition is mainly obtained by exploiting a reformulation of the problem based on the stability of a matrix over the complex unit disc, and by introducing suitable tables for establishing stability of polynomials with complex coefficients. The condition is sufficient for any size of the LMIs, and is also necessary for a size sufficiently large. Then, the paper proceeds by addressing a more general formulation, showing that the proposed approach can be used also when the plant is allowed to be nonstrictly proper, and when the output feedback controller is allowed to be dynamic.

Representation of free Herglotz functions

A Herglotz function is a holomorphic map from the open complex unit disk into the closed complex right halfplane. A classical Herglotz function has an integral representation against a positive measure on the unit circle. We prove a free analytic analogue of the Herglotz representation and describe how our representations specialize to the free probabilistic case. We also show that the set of representable Herglotz functions arising from noncommutative conditional expectations must be closed in a natural topology.

Analysis of a discrete-time single-server queue with an occasional extra server

We consider a discrete-time queueing system having two distinct servers: one server, the “regular” server, is permanently available, while the second server, referred to as the “extra” server, is only allocated to the system intermittently. Apart from their availability, the two servers are identical, in the sense that the customers have deterministic service times equal to 1 fixed-length time slot each, regardless of the server that processes them. In this paper, we assume that the extra server is available during random “up-periods”, whereas it is unavailable during random “down-periods”. Up-periods and down-periods occur alternately on the time axis. The up-periods have geometrically distributed lengths (expressed in time slots), whereas the distribution of the lengths of the down-periods is general, at least in the first instance. Customers enter the system according to a general independent arrival process, i.e., the numbers of arrivals during consecutive time slots are i.i.d. random variables with arbitrary distribution.For this queueing model, we are able to derive closed-form expressions for the steady-state probability generating functions (pgfs) and the expected values of the numbers of customers in the system at various observation epochs, such as the start of an up-period, the start of a down-period and the beginning of an arbitrary time slot. At first sight, these formulas, however, appear to contain an infinite number of unknown constants. One major issue of the mathematical analysis turns out to be the determination of these constants. In the paper, we show that restricting the pgf of the down-periods to be a rational function of its argument, brings about the crucial simplification that the original infinite number of unknown constants appearing in the formulas can be expressed in terms of a finite number of independent unknowns. The latter can then be adequately determined based on the bounded nature of pgfs inside the complex unit disk, and an extensive use of properties of polynomials.Various special cases, both from the perspective of the arrival distribution and the down-period distribution, are discussed. The results are also illustrated by means of relevant numerical examples.Possible applications of this type of queueing model are numerous: the extra server could be the regular server of another similar queue, helping whenever an idle period occurs in its own queue; a geometric distribution for these idle times is then a very natural modeling assumption. A typical example would be the situation at the check-in counter at a gate in an airport: the regular server serves customers with a low-fare ticket, while the extra server gives priority to the business-class and first-class customers, but helps checking regular customers, whenever the priority line is empty.

Landau’s theorem for slice regular functions on the quaternionic unit ball

During the development of the theory of slice regular functions over the real algebra of quaternions [Formula: see text] in the last decade, some natural questions arose about slice regular functions on the open unit ball [Formula: see text] in [Formula: see text]. This work establishes several new results in this context. Along with some useful estimates for slice regular self-maps of [Formula: see text] fixing the origin, it establishes two variants of the quaternionic Schwarz–Pick lemma, specialized to maps [Formula: see text] that are not injective. These results allow a full generalization to quaternions of two theorems proven by Landau for holomorphic self-maps [Formula: see text] of the complex unit disk with [Formula: see text]. Landau had computed, in terms of [Formula: see text], a radius [Formula: see text] such that [Formula: see text] is injective at least in the disk [Formula: see text] and such that the inclusion [Formula: see text] holds. The analogous result proven here for slice regular functions [Formula: see text] allows a new approach to the study of Bloch–Landau-type properties of slice regular functions [Formula: see text].

A uniform bound for the Lagrange polynomials of Leja points for the unit disk

We study uniform estimates for the family of fundamental Lagrange polynomials defined from any Leja sequence for the complex unit disk. The main result claims that all these polynomials are uniformly bounded on the disk, i.e. independently on the range $N$ of the associated $N$-Leja section. An essential and immediate application is an estimate of the associated Lebesgue constants as $\Lambda_N=O(N)$, $\forall\,N\geq1$. We also deal with the special case of $N=2^p-1$ where we give an improvement of the main result.