Unit ballB Research Articles

Surjective Nash maps between semialgebraic sets

In this work we study the existence of surjective Nash maps between two given semialgebraic sets S and T. Some key ingredients are: the irreducible components Si⁎ of S (and their intersections), the analytic path-connected components Tj of T (and their intersections) and the relations between dimensions of the semialgebraic sets Si⁎ and Tj. A first step to approach the previous problem is the former characterization done by the second author of the images of affine spaces under Nash maps.The core result of this article to obtain a criterion to decide about the existence of surjective Nash maps between two semialgebraic sets is: a full characterization of the semialgebraic subsetsS⊂Rnthat are the image of the closed unit ballB‾mofRmcentered at the origin under a Nash mapf:Rm→Rn. The necessary and sufficient conditions that must satisfy such a semialgebraic set S are: it is compact, connected by analytic paths and has dimensiond≤m.Two remarkable consequences of the latter result are the following: (1) pure dimensional compact irreducible arc-symmetric semialgebraic sets of dimension d are Nash images ofB‾d, and (2) compact semialgebraic sets of dimension d are projections of non-singular algebraic sets of dimension d whose connected components are Nash diffeomorphic to spheres (maybe of different dimensions).

Daugavet property of Banach algebras of holomorphic functions and norm-attaining holomorphic functions

We show that the duals of Banach algebras of scalar-valued bounded holomorphic functions on the open unit ball BE of a Banach space E lack weak⁎-strongly exposed points. Consequently, we obtain that some Banach algebras of holomorphic functions on an arbitrary Banach space have the Daugavet property which extends the observation of P. Wojtaszczyk [56]. Moreover, we present a new denseness result by proving that the set of norm-attaining vector-valued holomorphic functions on the open unit ball of a dual Banach space is dense provided that its predual space has the metric π-property. Besides, we obtain several equivalent statements for the Banach space of vector-valued homogeneous polynomials to be reflexive, which improves the result of J. Mujica [47], J. A. Jaramillo and L. A. Moraes [39]. As a byproduct, we generalize some results on polynomial reflexivity due to J. Farmer [35].

𝒩(p, q, s)-type spaces in the unit ball of ℂ n (II): Carleson measure and its application

Abstract The purpose of this paper is to study a new class of function spaces, called 𝒩 ⁢ ( p , q , s ) {\mathcal{N}(p,q,s)} -type spaces, in the unit ball 𝔹 {{\mathbb{B}}} of ℂ n {{\mathbb{C}}^{n}} . The Carleson measure on such spaces is investigated. Some embedding theorems among 𝒩 ⁢ ( p , q , s ) {\mathcal{N}(p,q,s)} -type spaces, weighted Bergman spaces and weighted Hardy spaces are established. As for applications, the Hadamard products and random power series on 𝒩 ⁢ ( p , q , s ) {\mathcal{N}(p,q,s)} -type spaces are also studied.

Einstein Gyrogroup as a B-loop

Using the Clifford algebra formalism, we give an algebraic proof that the open unit ball B = { v ∈ R n : ‖ v ‖ 1 } of R n equipped with Einstein addition ⊕E forms a B-loop or, equivalently, a uniquely 2-divisible gyrocommutative gyrogroup. We obtain a compact formula for Einstein addition in terms of Mobius addition. We then give a characterization of associativity and commutativity of vectors in B with respect to Einstein addition.

A note on monolithic scattered compacta

For a Banach space E, it is well-known that a necessary condition for E to have the controlled separable complementation property (CSCP, for short) is that the dual unit ball BE⁎ be monolithic in the weak-star topology. We prove here that when X is a scattered first countable locally compact space, then monolithicity of X turns out to be sufficient for C0(X) to enjoy the CSCP.

Spectra of weighted composition operators on algebras of analytic functions on banach spaces

Let E be a complex Banach space, with the unit ball BE. We study the spectrum of a bounded weighted composition operator uCφ on H∞(BE) determined by an analytic symbol φ with a fixed point in BE such that φ(BE) is a relatively compact subset of E, where u is an analytic function on BE.

Spectra of composition operators on algebras of analytic functions on Banach spaces

Let E be a Banach space, with unit ball BE. We study the spectrum and the essential spectrum of a composition operator on H∞(BE) determined by an analytic symbol with a fixed point in BE. We relate the spectrum of the composition operator to that of the derivative of the symbol at the fixed point. We extend a theorem of Zheng to the context of analytic symbols on the open unit ball of a Hilbert space.

Weighted Composition Operators fromF(p,q,s)Spaces toHμ∞Spaces

LetH(B)denote the space of all holomorphic functions on the unit ballB. Letu∈H(B)andφbe a holomorphic self-map ofB. In this paper, we investigate the boundedness and compactness of the weighted composition operatoruCφfrom the general function spaceF(p,q,s)to the weighted-type spaceHμ∞in the unit ball.

On reduced polytopes and antipodality

AbstractLetBbe ano-symmetric convex body in ℝd, andMdbe the normed space with unit ballB. TheMd-thickness ΔB(K) of a convex bodyK⊆ ℝdis the smallest possibleMd-distance between two distinct parallel supporting hyperplanes ofK. Furthermore,Kis said to beMd-reduced if ΔB(K′) < ΔB(K) for every convex bodyK′ withK′ ⊆KandK′ ≠K. In our main theorems we describeMd-reduced polytopes as polytopes whose face lattices possess certain antipodality properties. As one of the consequences, we obtain that if the boundary ofBis regular, then ad-polytope withmfacets andnvertices is notMd-reduced providedm=d+ 2 orn=d+ 2 orn>m. The latter statement yields a new partial answer to Lassak's question on the existence of Euclidean reducedd-polytopes ford≥ 3.

On a New Integral-Type Operator from the Weighted Bergman Space to the Bloch-Type Space on the Unit Ball

We introduce an integral-type operator, denoted byPφg, on the space of holomorphic functions on the unit ballB⊂ℂn, which is an extension of the product of composition and integral operators on the unit disk. The operator norm ofPφgfrom the weighted Bergman spaceAαp(B)to the Bloch-type spaceℬμ(B)or the little Bloch-type spaceℬμ,0(B)is calculated. The compactness of the operator is characterized in terms of inducing functionsgandφ. Upper and lower bounds for the essential norm of the operatorPφg:Aαp(B)→ℬμ(B), whenp>1, are also given.

Commutative C*-Algebras of Toeplitz Operators on the Unit Ball, I. Bargmann-Type Transforms and Spectral Representations of Toeplitz Operators

Extending known results for the unit disk, we prove that for the unit ballB n there exist n+2 difierent cases of commutative C ⁄ -algebras generated by Toeplitz operators, acting on weighted Bergman spaces. In all cases the bounded measurable symbols of Toeplitz operators are invariant under the action of certain commutative subgroups of biholomorphisms of the unit ball. Mathematics Subject Classiflcation (2000). Primary 47B35; Secondary 47L80, 32A36.

Concentrating solutions for the Hénon equation in ℝ2

We consider the boundary value problem Δu+⋎x⋎2α u p =0, α>0, in the unit ballB with homogeneous Dirichlet boundary condition andp a large exponent. We find a condition which ensures the existence of a positive solutionu p concentrating outside the origin atk symmetric points asp goes to +∞. The same techniques lead also to a more general result on general domains. In particular, we find that concentration at the origin is always possible, provided α⊄IN.

A generalization of half-plane mappings to the ball in ℂⁿ

Let F F be a normalized ( F ( 0 ) = 0 F(0)=0 , D F ( 0 ) = I DF(0)=I ) biholomorphic mapping of the unit ball B ⊆ C n B \subseteq \mathbb C^n onto a convex domain Ω ⊆ C n \Omega \subseteq \mathbb C^n that is the union of lines parallel to some unit vector u ∈ C n u \in \mathbb C^n . We consider the situation in which there is one infinite singularity of F F on ∂ B \partial B . In one case with a simple change-of-variables, we classify all convex mappings of B B that are half-plane mappings in the first coordinate. In the more complicated case, when u u is not in the span of the infinite singularity, we derive a form of the mappings in dimension n = 2 n=2 .

On the Inequality for Volume and Minkowskian Thickness

AbstractGiven a centrally symmetric convex bodyBin, we denote by ℳd(B) the Minkowski space (i.e., finite dimensional Banach space) with unit ballB. LetKbe an arbitrary convex body in ℳd(B). The relationship between volumeV(K) and the Minkowskian thickness (= minimal width) ΔB(K) ofKcan naturally be given by the sharp geometric inequalityV(K) ≥ α(B) · ΔB(K)d, where α(B) > 0. As a simple corollary of the Rogers-Shephard inequality we obtain thatwith equality on the left attained if and only ifBis the difference body of a simplex and on the right ifBis a cross-polytope. The main result of this paper is that ford= 2 the equality on the right implies thatBis a parallelogram. The obtained results yield the sharp upper bound for the modified Banach–Mazur distance to the regular hexagon.

Spectra of algebras of entire functions on Banach spaces

We obtain an explicit description of the spectrum (set of closed maximal ideals) of H b ( X ) , H_b(X), algebra of analytic functions on a Banach space X X which are bounded on bounded subsets. We show that the spectrum of H b ( X ) H_b(X) admits a natural linear structure. Some applications to the algebra of uniformly continuous and bounded analytic functions on the unit ball B ⊂ X B\subset X are indicated.

Quasiconformal extension of biholomorphic mappings in several complex variables

Letf(z, t) be a subordination chain fort ∈ [0, α], α>0, on the Euclidean unit ballB inC n. Assume thatf(z) =f(z, 0) is quasiconformal. In this paper, we give a sufficient condition forf to be extendible to a quasiconformal homeomorphism on a neighbourhood of\(\bar B\). We also show that, under this condition,f can be extended to a quasiconformal homeomorphism of\(\overline {R^{2n} } \) onto itself and give some applications.

An extremal function for the Chang-Marshall inequality over the Beurling functions

S.-Y. A. Chang and D. E. Marshall showed that the functional Λ ( f ) = ( 1 / 2 π ) ∫ 0 2 π exp ⁡ { | f ( e i θ ) | 2 } d θ \Lambda (f) =(1/2\pi ) \int _0^{2\pi }\exp \{ |f(e^{i\theta })|^2\}d\theta is bounded on the unit ball B \mathcal {B} of the space D \mathcal {D} of analytic functions in the unit disk with f ( 0 ) = 0 f(0)=0 and Dirichlet integral not exceeding one. Andreev and Matheson conjectured that the identity function f ( z ) = z f(z)=z is a global maximum on B \mathcal {B} for the functional Λ \Lambda . We prove that Λ \Lambda attains its maximum at f ( z ) = z f(z)=z over a subset of B \mathcal {B} determined by kernel functions, which provides a positive answer to a conjecture of Cima and Matheson.

Boundary behavior of Gleason's problem in hyperbolic harmonic Bergman spaces

Necessary and sufficient conditions are obtained for the boundedness of Berezin transformation on Lebesgue space Lp(B, dVβ) in the real unit ballB in ℝn. As an application, we prove that Gleason type problem is solvable in hyperbolic harmonic Bergman spaces. Furthermore we investigate the boundary behavior of the solutions of Gleason type problem.

BOUNDARIES FOR AN ALGEBRA OF BOUNDED HOLOMORPHIC FUNCTIONS

Let Ab(BE) be the Banach algebra of all complex val- ued bounded continuous functions on the closed unit ball BE of a complex Banach space E, and holomorphic in the interior of BE, endowed with the sup norm. We present some sucient conditions for a set to be a boundary for Ab(BE) in case E belongs to a class of Banach spaces that includes the pre-dual of a Lorentz sequence space studied by Gowers in (6). We also prove the non-existence of the Shilov boundary for Ab(BE) and give some examples of bound- aries.

Connected components in the space of composition operators onH∞ functions of many variables

LetE be a complex Banach space with open unit ballBe. The structure of the space of composition operators on the Banach algebra H∞, of bounded analytic functions onBe with the uniform topology, is studied. We prove that the composition operators arising from mappings whose range lies strictly insideBe form a path connected component. WhenE is a Hilbert space or aCo(X)- space, the path connected components are shown to be the open balls of radius 2.