Spectral and dynamical results related to certain non-integer base expansions on the unit interval

Operator-valued multivariable Bohr type inequalities are obtained for: [(i)] a class of noncommutative holomorphic functions on the open unit ball of $B(\mathcal {H})^n$, generalizing the analytic functions on the open unit disc; [(ii)] the noncommutative disc algebra $\mathcal {A}_n$ and the noncommutative analytic Toeplitz algebra $F_n^\infty$; [(iii)] a class of noncommutative selfadjoint harmonic functions on the open unit ball of $B(\mathcal {H})^n$, generalizing the real-valued harmonic functions on the open unit disc; [(iv)] the Cuntz-Toeplitz algebra $C^*(S_1,\ldots , S_n)$, the reduced (resp. full) group $C^*$-algebra $C_{red}^*(\mathbb {F}_n)$ (resp. $C^*(\mathbb {F}_n)$) of the free group with $n$ generators; [(v)] a class of analytic functions on the open unit ball of $\mathbb {C}^n$. The classical Bohr inequality is shown to be a consequence of Fejér's inequality for the coefficients of positive trigonometric polynomials and Haager- up-de la Harpe inequality for nilpotent operators. Moreover, we provide an inequality which, for analytic polynomials on the open unit disc, is sharper than Bohr's inequality.

In this paper, expansive mappings are seen as generalizations of expansive homeomorphisms, and known theorems are generalized. The nonexistence of expansive functions on the open unit interval is proven, and a new example of an expansive homeomorphism on the open unit disk is given. Finally, a method for constructing expansive homeomorphisms from expansive mappings is given.

We study a variant of intersection representations with unit balls: unit disks in the plane and unit intervals on the line. Given a planar graph and a bipartition of the edges of the graph into near and far edges, the goal is to represent the vertices of the graph by unit-size balls so that the balls for two adjacent vertices intersect if and only if the corresponding edge is near. We consider the problem in the plane and prove that it is NP-hard to decide whether such a representation exists for a given edge-partition. On the other hand, we show that series-parallel graphs which include outerplanar graphs admit such a representation with unit disks for any near/far bipartition of the edges. The unit-interval on the line variant is equivalent to threshold graph coloring, in which context it is known that there exist girth-3 planar graphs even outerplanar graphs that do not admit such coloring. We extend this result to girth-4 planar graphs. On the other hand, we show that all triangle-free outerplanar graphs and all planar graphs with maximum average degree less than 26/11 have such a coloring, via unit-interval intersection representation on the line. This gives a simple proof that all planar graphs with girth at least 13 have a unit-interval intersection representation on the line.

We define the Dirichlet space 31 on the unit polydisc of . 2 is a semi-Hilbert space of of holomorphic functions, contains the holomorphic polynomials densely, is invariant under compositions with the biholomorphic automorphisms of , and its semi-norm is preserved under such compositions. We show that 2 is unique with these properties. We also prove 3ί is unique if we assume that the semi-norm of a function in 3f composed with an automorphism is only equivalent in the metric sense to the semi-norm of the original function. Members of a subclass of 3f given by a norm can be written as potentials of «5 5>2-functions on the «-torus . We prove that the functions in this subclass satisfy strong-type inequalities and have tangential limits almost everywhere on dVn . We also make capacitory estimates on the size of the exceptional sets on f)Vn . 1. Introduction. Mόbius-invariant spaces. Let U be the open unit disc in C and T be the unit circle bounding it. The open unit polydisc Vn and the torus Tn in Cn are the cartesian products of n unit discs and n unit circles, respectively. Ίn is the distinguished boundary of Uw and forms only a small part of the topological boundary dVn of Un . We denote by Jΐ the group of all biholomorphic automorphisms of Vn (the Mδbius group). The subgroup of linear automorphisms in Jf is denoted by %. The space of holomorphic functions with domain Un will be called ^(ϋn) and will carry the topology of uniform convergence on compact subsets of Vn. A semi-inner product on a complex vector space β? is a sesquilinear functional on β(? x %? with all the properties of an inner product except that it is possible to have {(a, a)) = 0 when a Φ 0. IWI = \/(( α> a)) *s the associated semi-norm. We assume ((• , •)) is

In this effort, we employ some of the linear differential inequalities to achieve integral inequalities of the type Wiener–Hopf problems (WHP). We utilize the concept of subordination and its applications to gain linear integral operators in the open unit disk that preserve two classes of analytic functions with a positive real part. Linear second-order differential inequalities play a significant role in the field of complex differential equations. Our study is based on a neighborhood containing the origin. Therefore, the Wiener–Hopf problem is decomposed around the origin in the open unit disk using two different classes of analytic functions. Moreover, we suggest a generalization for WHP by utilizing some classes of entire functions. Special cases are given in the sequel as well. A necessary and sufficient condition for WHP to be averaging operator on a convex domain (in the open unit disk) is given by employing the subordination relation (inequality).

Suitability in fuzzy topological spaces

We consider a generalization of the well-known problem to randomly fill a long segment by unit intervals. On the segment [0, x], x ≥ 1, we place an open unit interval according to the Fx distribution law, which is the distribution of the left-hand endpoint of the unit interval, concentrated on the segment [0, x – 1]. Let the first allocated interval (t, t + 1) divide the segment [0, x] into two parts [0, t] and [t + 1, x] and they are filled independently of each other according to the following rules. On the segment [0, t] a point t1 is selected randomly according to the law Ft and the interval (t1, t1 + 1) is placed. A point t2 is selected randomly in the segment [t + 1, x] such that u = t2 – t – 1 is a random variable distributed according to the law Fx – t – 1, and we place the interval (t2, t2 + 1). In the same way, the newly formed segments are then filled. If x < 1, then the filling process is considered to be complete and the unit interval is not placed on the segment [0, x]. At the end of the filling process, unit intervals are located on the segment [0, x] such that the distances between adjacent intervals are less than one. In this article, we consider distribution laws Fx with distribution densities such that their graphs are centrally symmetric with respect to the point (x – 1/2, 1/x – 1). In particular, this class of distributions includes the uniform distribution on the segment [0, x – 1] (the corresponding filling problem was previously investigated by other authors). Let Nx be the total amount of single units placed on the segment [0, x]. Our concern is the properties of the distribution of this random variable. We obtain an asymptotic description of the behavior of central moments and prove the asymptotic normality of the random variable Nx. In addition, we establish that the distributions of the random variables Nx are the same for all the distribution laws of the specified class.

Operators affiliated to the free shift on the Free Hardy Space

In this paper, we establish a Schwarz–Pick lemma for the norms of holomorphic mappings defined in open unit balls of JB-triples. In addition, the extremal problems are studied for holomorphic mappings from the open unit disk in to the open unit balls of the complex Banach spaces or Hilbert spaces.

The following extension theorem is proved. Let $\Omega \subset {\mathbf {C}}$ be an open set containing $\Delta$, the open unit disc in ${\mathbf {C}}$, and the point 1. Suppose that $f$ is holomorphic on $B$, the open unit ball of ${{\mathbf {C}}^N}$, let $x \in \partial B$ and assume that for all $y \in \partial B$ in a neighborhood of $x$ the function $\varsigma \to f(\varsigma y)$, holomorphic on $\Delta$, continues analytically into $\Omega$. Then $f$ continues analytically into a neighborhood of $x$.

Bounded symmetric domains (Cartan domains and exceptional domains) are higher-dimensional generalizations of the open unit disc. In this note we give a structure theory for the C*-algebra T generated by all Toeplitz operators Tf(h) := P{fh) with continuous symbol function ƒ G C(S) on the Shilov boundary 5 of a bounded symmetric domain D of arbitrary rank r. Here h belongs to the Hardy space H(S), and P : L(S) —• H(S) is the Szego projection. For domains of rank 1 and tube domains of rank 2, the structure of T has been determined in [1, 2]. In these cases Toeplitz operators are closely related to pseudodifferential operators. For the open unit disc, T is the C*-algebra generated by the unilateral shift. The structure theory for the general case [12] is based on the fact that D can be realized as the open unit ball of a unique Jordan triple system Z [7, Theorem 4.1]. Denoting the Jordan triple product by {uv*w}, a tripotent e G Z satisfies {ee*e} = e. Tripotents generalize the partial isometries of matrix algebras and determine the boundary structure of D C Z (cf. [7, Theorem 6.3]). Our principal result ([12]; cf. also [3, 4, 8]) is the following;

Meromorphic interpolation in several variables

In this paper we continue the study of free holomorphic functions on the noncommutative ball where B (ℋ) is the algebra of all bounded linear operators on a Hilbert space ℋ, and n = 1, 2, . . . or n = ∞. These are noncommutative multivariable analogues of the analytic functions on the open unit disc 𝔻 ≔ { z ∈ ℂ : | z | &lt; 1}. The theory of characteristic functions for row contractions (elements in ) is used to determine the group of all free holomorphic automorphisms of [ B (ℋ) n ] 1 . It is shown that , the Moebius group of the open unit ball 𝔹 n ≔ { λ ∈ ℂ n : ∥ λ ∥ 2 &lt; 1}. We show that the noncommutative Poisson transform commutes with the action of the automorphism group . This leads to a characterization of the unitarily implemented automorphisms of the Cuntz-Toeplitz algebra C *( S 1 , . . . , S n ), which leave invariant the noncommutative disc algebra . This result provides new insight into Voiculescu's group of automorphisms of the Cuntz-Toeplitz algebra and reveals new connections with noncommutative multivariable operator theory, especially, the theory of characteristic functions for row contractions and the noncommutative Poisson transforms. We show that the unitarily implemented automorphisms of the noncommutative disc algebra and the noncommutative analytic Toeplitz algebra , respectively, are determined by the free holomorphic automorphisms of [ B (ℋ) n ] 1 , via the noncommutative Poisson transform. Moreover, we prove that . We also prove that any completely isometric automorphism of the noncommutative disc algebra has the form where . We deduce a similar result for the noncommutative Hardy algebra , which, due to Davidson-Pitts results on the automorphisms of , implies that is isomorphic to the group of contractive automorphisms of . We study the isometric dilations and the characteristic functions of row contractions under the action of the automorphism group . This enables us to obtain some results concerning the behavior of the curvature and the Euler characteristic of a row contraction under . Finally, in the last section, we prove a maximum principle for free holomorphic functions on the noncommutative ball [ B (ℋ) n ] 1 and provide some extensions of the classical Schwarz lemma to our noncommutative setting.

In this paper we obtain a noncommutative multivariable analogue of the classical Nevanlinna–Pick interpolation problem for analytic functions with positive real parts on the open unit disc. Given a function $$f : \Lambda \to \mathbb {C}$$ , where $$\Lambda$$ is an arbitrary subset of the open unit ball $$\mathbb{B}_n:=\{z\in \mathbb {C}^n: \|z\| < 1\}$$ , we find necessary and sufficient conditions for the existence of a free holomorphic function g with complex coefficients on the noncommutative open unit ball $$[B({\mathcal H})^n]_1$$ such that $${\rm Re} \ g \geq 0 \quad {\rm and} \quad g(z)=f(z),\quad z\in \Lambda,$$ where $$B({\mathcal H})$$ is the algebra of all bounded linear operators on a Hilbert space $${\mathcal H}$$ . The proof employs several results from noncommutative multivariable operator theory and a noncommutative Cayley transform (introduced and studied in the present paper) acting from the set of all free holomorphic functions with positive real parts to the set of all bounded free holomorphic functions. All the results of this paper are obtained in the more general setting of free holomorphic functions with operator-valued coefficients. As consequences, we deduce some results concerning operator-valued analytic interpolation on the unit ball $${\mathbb B}_n$$ .

Abstract. We introduce a subclass S (k)s (A,B) (−1 ≤ B < A ≤ 1)of functions which are analytic in the open unit disk and close-to-convexwith respect to k-symmetric points. We give some coefficient inequalities,integral representations and invariance properties of functions belongingto this class. 1. IntroductionLet A denote the class of functions which are analytic in the open unit diskUand normalized by f(0) = 0 and f ′ (0) = 1. Also let S denote the subclassof A consisting of all functions which are univalent in U.Let f(z) and F(z) be analytic in U. Then we say that the function f(z) issubordinate to F(z) in U, if there exists an analytic function w(z) in U suchthat |w(z)| ≤ 1 and f(z) = F(w(z)), denote by f ≺ F or f(z) ≺ F(z). IfF(z) is univalent in U, then the subordination is equivalent to f(0) = F(0) andf(U) ⊂ F(U).Now, we denote by S ∗ (A,B) and C(A,B) the subclasses of A as follows:(1) S ∗ (A,B) =ˆf ∈ A :zf ′ (z)f(z)≺1+Az1+Bz,z ∈ U˙and(2) C(A,B) =ˆf ∈ A : ∃g ∈ S ∗ (A,B) such thatzf