Unit Polydisc Research Articles

On the classification of function algebras on subvarieties of noncommutative operator balls

We study algebras of bounded noncommutative (nc) functions on unit balls of operator spaces (nc operator balls) and on their subvarieties. Considering the example of the nc unit polydisk we show that these algebras, while having a natural operator algebra structure, might not be the multiplier algebra of any reasonable nc reproducing kernel Hilbert space (RKHS). After examining additional subtleties of the nc RKHS approach, we turn to study the structure and representation theory of these algebras using function theoretic and operator algebraic tools. We show that the underlying nc variety is a complete invariant for the algebra of uniformly continuous nc functions on a homogeneous subvariety, in the sense that two such algebras are completely isometrically isomorphic if and only if the subvarieties are nc biholomorphic. We obtain extension and rigidity results for nc maps between subvarieties of nc operator balls corresponding to injective spaces that imply that a biholomorphism between homogeneous varieties extends to a biholomorphism between the ambient balls, which can be modified to a linear isomorphism. Thus, the algebra of uniformly continuous nc functions on nc operator balls, and even its restriction to certain subvarieties, completely determine the operator space up to completely isometric isomorphism.

Some Results on Composition of Analytic Functions in a Unit Polydisc

Some Results on Composition of Analytic Functions in a Unit Polydisc

Clark Measures on Polydiscs Associated to Product Functions and Multiplicative Embeddings

We study Clark measures on the unit polydisc, giving an overview of recent research and investigating the Clark measures of some new examples of multivariate inner functions. In particular, we study the relationship between Clark measures and multiplication; first by introducing compositions of inner functions and multiplicative embeddings, and then by studying products of one-variable inner functions.

Analytic Functions in a Complete Reinhardt Domain Having Bounded L-Index in Joint Variables

The manuscript is an initiative to construct a full and exhaustive theory of analytical multivariate functions in any complete Reinhardt domain by introducing the concept of L-index in joint variables for these functions for a given continuous, non-negative, non-vanishing, vector-valued mapping L defined in an interior of the domain with some behavior restrictions. The complete Reinhardt domain is an example of a domain having a circular symmetry in each complex dimension. Our results are based on the results obtained for such classes of holomorphic functions: entire multivariate functions, as well as functions which are analytical in the unit ball, in the unit polydisc, and in the Cartesian product of the complex plane and the unit disc. For a full exhaustion of the domain, polydiscs with some radii and centers are used. Estimates of the maximum modulus for partial derivatives of the functions belonging to the class are presented. The maximum is evaluated at the skeleton of some polydiscs with any center and with some radii depending on the center and the function L and, at most, it equals a some constant multiplied by the partial derivative modulus at the center of the polydisc. Other obtained statements are similar to the described one.

Sharp inequalities for holomorphic function spaces

In this paper we establish some sharp inequalities for holomorphic Fock spaces in Cn and weighted Bergman spaces in the unit polydisk, which generalizes some results of Burbea. As an application, we prove Furdui's conjecture about the norm of composition operators between Fock spaces.

Multiplication Operators on Weighted Zygmund Spaces of the First Cartan Domain

Inspired by some recent studies of the multiplication operators on holomorphic function spaces of the classical domains such as the open unit disk, the unit ball and the unit polydisk, the purpose of the present paper is to study just the operators that are defined on weighted Zygmund spaces of the first Cartan domain. We obtain some necessary conditions and sufficient conditions for the operators to be bounded and compact.

ON POSITIVE CONTINUOUS FUNCTIONS DEFINED IN THE UNIT POLYDISC

In theory of holomorphic functions having bounded L-index in a direction b an auxiliary class of positive continuous functions L is important to describe properties of the holomorphic functions by some inequalities andestimates containing the function L. This class is defined by local behavior of the function L. In the simplest one-dimensional case, the functionshould not vary locally too quickly, i.e. L(r+O(1/L(r))) = O(L(r)) for r = |z| → +∞. The paper is devoted an analog of this function class for the unit polydisc, i.e. for the Cartesian product of the unit discs. There is proved an equivalence of three different approaches to define the class. It is described by the local behavior on the slice z+tb for given z fromthe unit polydisc and for a fixed direction b, where the complex variable t belongs to some disc with radius dependent on b and z. These estimates must be fulfilled uniformly in all z. There is indicated a possible explicitform of functions belonging to the class. The form is given as a product of arbitrary positive continuous function defined in the closed unit polydisc and the minimum of the expressions 1/(1−|z j|) in all variables z j.

Pairs of inner projections and two applications

Orthogonal projections onto closed subspaces of H2(Dn) of the form φH2(Dn) for inner functions φ on Dn are referred to as inner projections, where H2(Dn) denotes the Hardy space over the open unit polydisc Dn. In this paper, we classify pairs of commuting inner projections. We also present two seemingly independent applications: the first is an answer to a question posed by R. G. Douglas, and the second is a complete classification of partially isometric truncated Toeplitz operators with inner symbols on Dn.

Analytic in the unit polydisc functions of bounded L-index in direction

The concept of bounded $L$-index in a direction $\mathbf{b}=(b_1,\ldots,b_n)\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ is generalized for a class of analytic functions in the unit polydisc, where $L$ is some continuous function such that for every $z=(z_1,\ldots,z_n)\in\mathbb{D}^n$ one has $L(z)>\beta\max_{1\le j\le n}\frac{|b_j|}{1-|z_j|},$ $\beta=\mathrm{const}>1,$ $\mathbb{D}^n$ is the unit polydisc, i.e. $\mathbb{D}^n=\{z\in\mathbb{C}^n: |z_j|\le 1, j\in\{1,\ldots,n\}\}.$ For functions from this class we obtain sufficient and necessary conditions providing boundedness of $L$-index in the direction. They describe local behavior of maximum modulus of derivatives for the analytic function $F$ on every slice circle $\{z+t\mathbf{b}: |t|=r/L(z)\}$ by their values at the center of the circle, where $t\in\mathbb{C}.$ Other criterion describes similar local behavior of the minimum modulus via the maximum modulus for these functions. We proved an analog of the logarithmic criterion desribing estimate of logarithmic derivative outside some exceptional set by the function $L$. The set is generated by the union of all slice discs $\{z^0+t\mathbf{b}: |t|\le r/L(z^0)\}$, where $z^0$ is a zero point of the function $F$. The analog also indicates the zero distribution of the function $F$ is uniform over all slice discs. In one-dimensional case, the assertion has many applications to analytic theory of differential equations and infinite products, i.e. the Blaschke product, Naftalevich-Tsuji product. Analog of Hayman's Theorem is also deduced for the analytic functions in the unit polydisc. It indicates that in the definition of bounded $L$-index in direction it is possible to remove the factorials in the denominators. This allows to investigate properties of analytic solutions of directional differential equations.

Extension operators and support points associated with g-Loewner chains on complex Banach spaces

We prove that the Roper-Suffridge extension operator preserves g-parametric representation on general domains Bp1,p2={(z1,w)∈C×X:|z1|p1+‖w‖Xp2<1}, where X is a complex Banach space and p1,p2≥1. This result generalizes recent result from p1=2 to 1≤p1<∞. Next, we obtain a rigidity property of the Roper-Suffridge extension operator for the preservation of convexity. Finally, some sharp coefficient bounds for univalent mappings with g-parametric representation on the unit p-ball of Cn (1<p≤∞) are also obtained. As applications, we establish bounded support points with g-parametric representation on the unit p-ball. These results cover recent results related to bounded support points for families of biholomorphic mappings on the Euclidean unit ball for p=2, and on the unit polydisc for p=∞, respectively.

Coefficient bounds for class of g-starlike mappings of complex order γ in

In this paper, the definition of g-starlike mappings of complex order γ is extended from the case of unit open disk in C to the cases of the unit ball in a complex Banach space, unit polydisk and bounded starlike circular domain in C n , respectively, where g are the Carathéodory functions and γ ∈ C ∖ { 0 } . The sharp unified solutions of Fekete–Szegö type problems for g-starlike mappings of complex order γ in several complex variables are solved. Compare with some recent corresponding works, the main results hold true without any restrictive assumptions and the proofs are also more simple. All theorems can naturally be reduced to the different cases of subclasses of starlike mappings and g-starlike mappings by choosing suitable γ and g.

Complex symmetric Toeplitz operators on the unit polydisk

In this paper, we study the complex symmetry of Toeplitz operators on the weighted Bergman spaces over the unit polydisk. First, we completely characterize when anti-linear weighted composition operators [Formula: see text] are conjugations. We then give a sufficient and necessary condition for Toeplitz operators to be complex symmetric with respect to these conjugations. As a consequence, some interesting higher-dimensional complex symmetric phenomena appear on the unit polydisk such as the monomial Toeplitz operators [Formula: see text] with [Formula: see text] for some symmetric permutation matrix [Formula: see text]. Surprisingly, these operators [Formula: see text] are the only ones that are complex symmetric monomial Toeplitz operators on the unit bidisk.

О сопряженном к пространству функций, голоморфных на замкнутом полидиске, с мультипликативной сверткой

The algebra of analytic functionals on the closed unit polydisk 𝑫𝟏 ̅̅̅̅ in ℂ𝑵 is studied. The Hadamard type operators are characterized in the space 𝑯(𝑫𝟏 ̅̅̅̅) of all germs of holomorphic on 𝑫𝟏 ̅̅̅̅ functions. We obtain the representation of mentioned algebra in 𝑯(𝑫𝟏 ̅̅̅̅) as the Hadamard type operators algebra. The multiplicative functionals on it, its Jacobson radical and idempotents are described.

A note on polydegree (n, 1) rational inner functions, slice matrices, and singularities

We analyze certain compositions of rational inner functions in the unit polydisk mathbb {D}^{d} with polydegree (n, 1), nin mathbb {N}^{d-1}, and isolated singularities in mathbb {T}^d. Provided an irreducibility condition is met, such a composition is shown to be a rational inner function with singularities in precisely the same location as those of the initial function, and with quantitatively controlled properties. As an application, we answer a d-dimensional version of a question posed in Bickel et al. (Am J Math 144: 1115–1157, 2022) in the affirmative.

A Note on Pluriharmonic Functions in the Unit Polydisc in ℂn

A Note on Pluriharmonic Functions in the Unit Polydisc in ℂn

Morera’s Boundary Theorem in Siegel Domain of the First Kind

The paper considers the realization of the matrix unit polydisk in the form of a Siegel domain of the first kind and proves the boundary analogue of Morera’s theorem

A rigidity theorem at the boundary for holomorphic mappings with values in finite dimensional bounded symmetric domains

AbstractLet be a bounded symmetric domain realized as the open unit ball of a finite dimensional JB*‐triple X. In this paper, we obtain a rigidity theorem at the boundary for holomorphic mappings from a balanced domain G in a complex Banach space E into . We also obtain a rigidity theorem at the boundary for holomorphic self‐mappings of . Our results give generalizations of the recent results obtained on the Euclidean unit ball or the unit polydisc in .

Some sharp Schwarz-Pick type estimates and their applications of harmonic and pluriharmonic functions

The purpose of this paper is to study the Schwarz-Pick type inequalities for harmonic or pluriharmonic functions. By analogy with the generalized Khavinson conjecture, we first give some sharp estimates of the norm of harmonic functions from the Euclidean unit ball in Rn into the unit ball of the real Minkowski space. Next, we give several sharp Schwarz-Pick type inequalities for pluriharmonic functions from the Euclidean unit ball in Cn or from the unit polydisc in Cn into the unit ball of the Minkowski space. Furthermore, we establish some sharp coefficient type Schwarz-Pick inequalities for pluriharmonic functions defined in the Minkowski space. Finally, we use the obtained Schwarz-Pick type inequalities to discuss the Lipschitz continuity, the Schwarz-Pick type lemmas of arbitrary order and the Bohr phenomenon of harmonic or pluriharmonic functions.

Characterizations of Herglotz\u2013Nevanlinna Functions Using Positive Semi-Definite Functions and the Nevanlinna Kernel in Several Variables

In this paper, we give several characterizations of Herglotz–Nevanlinna functions in terms of a specific type of positive semi-definite functions called Poisson-type functions. This allows us to propose a multidimensional analogue of the classical Nevanlinna kernel and a definition of generalized Nevanlinna functions in several variables. Furthermore, a characterization of the symmetric extension of a Herglotz–Nevanlinna function is also given. The subclass of Loewner functions is discussed as well, along with an interpretation of the main result in terms of holomorphic functions on the unit polydisk with non-negative real part.

The Fekete and Szegö inequality for a class of holomorphic mappings on the unit polydisk in n?> and its application

Let be the familiar class of normalized close-to-convex functions in the unit disk. In Koepf [On the Fekete-Szegö problem for close-to-convex functions. Proc Amer Math Soc. 1987;101:89–95], Koepf proved that for a function in the class , As an important application, in the same paper, Koepf showed that for close-to-convex functions. In this paper, we extend the above results to a subclass of close-to-quasi-convex mappings of type B defined on the unit polydisc in , and establish the sharp difference bound for the second and third coefficients of homogeneous expansions for this class of holomorphic mappings. The results presented here would provide a new path for solving the Bieberbach conjectures in several complex variables.