scholarly journals On the relation of Carleson's embedding and the maximal theorem in the context of Banach space geometry

2011 ◽  
Vol 109 (2) ◽  
pp. 269 ◽  
Author(s):  
Tuomas Hytönen ◽  
Mikko Kemppainen
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Maximal Operator ◽  
Embedding Theorem ◽  
The Other ◽  
Embedding Theorems ◽  
Geometric Properties ◽  
Maximal Theorem ◽  
Vector Valued ◽  
Funct Anal

Hytönen, McIntosh and Portal (J. Funct. Anal., 2008) proved two vector-valued generalizations of the classical Carleson embedding theorem, both of them requiring the boundedness of a new vector-valued maximal operator, and the other one also the type $p$ property of the underlying Banach space as an assumption. We show that these conditions are also necessary for the respective embedding theorems, thereby obtaining new equivalences between analytic and geometric properties of Banach spaces.

scholarly journals EQUIVALENT NORMS WITH AN EXTREMELY NONLINEABLE SET OF NORM ATTAINING FUNCTIONALS

2018 ◽  
Vol 19 (1) ◽  
pp. 259-279 ◽  
Author(s):  
Vladimir Kadets ◽  
Ginés López ◽  
Miguel Martín ◽  
Dirk Werner
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Finite Codimension ◽  
Countable System ◽  
Two Dimensional ◽  
Geometric Properties ◽  
Equivalent Norms ◽  
Norm Attaining ◽  
Original Norm ◽  
Funct Anal

We present a construction that enables one to find Banach spaces$X$whose sets$\operatorname{NA}(X)$of norm attaining functionals do not contain two-dimensional subspaces and such that, consequently,$X$does not contain proximinal subspaces of finite codimension greater than one, extending the results recently provided by Read [Banach spaces with no proximinal subspaces of codimension 2,Israel J. Math.(to appear)] and Rmoutil [Norm-attaining functionals need not contain 2-dimensional subspaces,J. Funct. Anal. 272(2017), 918–928]. Roughly speaking, we construct an equivalent renorming with the requested properties for every Banach space$X$where the set$\operatorname{NA}(X)$for the original norm is not “too large”. The construction can be applied to every Banach space containing$c_{0}$and having a countable system of norming functionals, in particular, to separable Banach spaces containing$c_{0}$. We also provide some geometric properties of the norms we have constructed.

scholarly journals Geometric Invariants of Surjective Isometries between Unit Spheres

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2346
Author(s):  
Almudena Campos-Jiménez ◽  
Francisco Javier García-Pacheco
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Geometric Property ◽  
The Other ◽  
Geometric Invariants ◽  
Finite Dimensional ◽  
Surjective Isometry ◽  
Finite Dimensional Case

In this paper we provide new geometric invariants of surjective isometries between unit spheres of Banach spaces. Let X,Y be Banach spaces and let T:SX→SY be a surjective isometry. The most relevant geometric invariants under surjective isometries such as T are known to be the starlike sets, the maximal faces of the unit ball, and the antipodal points (in the finite-dimensional case). Here, new geometric invariants are found, such as almost flat sets, flat sets, starlike compatible sets, and starlike generated sets. Also, in this work, it is proved that if F is a maximal face of the unit ball containing inner points, then T(−F)=−T(F). We also show that if [x,y] is a non-trivial segment contained in the unit sphere such that T([x,y]) is convex, then T is affine on [x,y]. As a consequence, T is affine on every segment that is a maximal face. On the other hand, we introduce a new geometric property called property P, which states that every face of the unit ball is the intersection of all maximal faces containing it. This property has turned out to be, in a implicit way, a very useful tool to show that many Banach spaces enjoy the Mazur-Ulam property. Following this line, in this manuscript it is proved that every reflexive or separable Banach space with dimension greater than or equal to 2 can be equivalently renormed to fail property P.

scholarly journals Multipliers on Vector Valued Bergman Spaces

2002 ◽  
Vol 54 (6) ◽  
pp. 1165-1186 ◽  
Author(s):  
Oscar Blasco ◽  
José Luis Arregui
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Bergman Space ◽  
Unit Disc ◽  
Bergman Spaces ◽  
Complex Banach Space ◽  
Bounded Operators ◽  
Taylor Coefficients ◽  
Vector Valued

AbstractLet X be a complex Banach space and let Bp(X) denote the vector-valued Bergman space on the unit disc for 1 ≤ p < ∞. A sequence (Tn)n of bounded operators between two Banach spaces X and Y defines a multiplier between Bp(X) and Bq(Y) (resp. Bp(X) and lq(Y)) if for any function we have that belongs to Bq(Y) (resp. (Tn(xn))n ∈ lq(Y)). Several results on these multipliers are obtained, some of them depending upon the Fourier or Rademacher type of the spaces X and Y. New properties defined by the vector-valued version of certain inequalities for Taylor coefficients of functions in Bp(X) are introduced.

An Embedding Theorem for Separable Locally Convex Spaces

1971 ◽  
Vol 14 (1) ◽  
pp. 119-120 ◽  
Author(s):  
Robert H. Lohman
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Space ◽  
Separable Banach Space ◽  
Locally Convex Space ◽  
Embedding Theorem ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Topological Vector Spaces ◽  
Locally Convex

A well-known embedding theorem of Banach and Mazur [1, p. 185] states that every separable Banach space is isometrically isomorphic to a subspace of C[0, 1], establishing C[0, 1] as a universal separable Banach space. The embedding theorem one encounters in a course in topological vector spaces states that every Hausdorff locally convex space (l.c.s.) is topologically isomorphic to a subspace of a product of Banach spaces.

scholarly journals The () Property in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Danyal Soybaş
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Bounded Linear Operator ◽  
Isomorphic Copy ◽  
Weakly Compact ◽  
Geometric Properties ◽  
Bounded Linear ◽  
Real Functions

A Banach space is said to have (D) property if every bounded linear operator is weakly compact for every Banach space whose dual does not contain an isomorphic copy of . Studying this property in connection with other geometric properties, we show that every Banach space whose dual has (V∗) property of Pełczyński (and hence every Banach space with (V) property) has (D) property. We show that the space of real functions, which are integrable with respect to a measure with values in a Banach space , has (D) property. We give some other results concerning Banach spaces with (D) property.

scholarly journals AN OPERATOR SUMMABILITY OF SEQUENCES IN BANACH SPACES

2013 ◽  
Vol 56 (2) ◽  
pp. 427-437 ◽  
Author(s):  
ANIL KUMAR KARN ◽  
DEBA PRASAD SINHA
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Borel Measure ◽  
Operator Ideal ◽  
The Other ◽  
Summable Sequence ◽  
Other Hand ◽  
Limited Operator

AbstractLet 1 ≤ p < ∞. A sequence 〈 xn 〉 in a Banach space X is defined to be p-operator summable if for each 〈 fn 〉 ∈ lw*p(X*) we have 〈〈 fn(xk)〉k〉n ∈ lsp(lp). Every norm p-summable sequence in a Banach space is operator p-summable whereas in its turn every operator p-summable sequence is weakly p-summable. An operator T ∈ B(X, Y) is said to be p-limited if for every 〈 xn 〉 ∈ lpw(X), 〈 Txn 〉 is operator p-summable. The set of all p-limited operators forms a normed operator ideal. It is shown that every weakly p-summable sequence in X is operator p-summable if and only if every operator T ∈ B(X, lp) is p-absolutely summing. On the other hand, every operator p-summable sequence in X is norm p-summable if and only if every p-limited operator in B(lp', X) is absolutely p-summing. Moreover, this is the case if and only if X is a subspace of Lp(μ) for some Borel measure μ.

scholarly journals On geometric properties of ⊥BJCϵ symmetric in some types of Banach spaces

2020 ◽  
Vol 1664 (1) ◽  
pp. 012038
Author(s):  
Saied A. Jhonny ◽  
Buthainah A. A. Ahmed
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Real Banach Space ◽  
Convex Banach Space ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Geometric Properties ◽  
Strictly Convex ◽  
Strictly Convex Banach Space

Abstract In this paper, we ⊥ B J C ϵ -orthogonality and explore ⊥ B J C ϵ -symmetricity such as a ⊥ B J C ϵ -left-symmetric ( ⊥ B J C ϵ -right-symmetric) of a vector x in a real Banach space (𝕏, ‖·‖𝕩) and study the relation between a ⊥ B J C ϵ -right-symmetric ( ⊥ B J C ϵ -left-symmetric) in ℐ(x). New results and proofs are include the notion of norm attainment set of a continuous linear functionals on a reflexive and strictly convex Banach space and using these results to characterize a smoothness of a vector in a unit sphere.

On the Duals of Flat Banach Spaces

1977 ◽  
Vol 20 (3) ◽  
pp. 293-299 ◽  
Author(s):  
Abraham Bick
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Dual Space ◽  
Geometric Properties

AbstractWe give a simpler proof to a theorem of L. A. Karlovitz that the dual of a flat Banach space is flat, and also study some geometric properties of the dual space.

Uniform Kadec–Klee–Huff properties of vector-valued Hardy spaces

1993 ◽  
Vol 114 (1) ◽  
pp. 25-30 ◽  
Author(s):  
P. N. Dowling ◽  
C. J. Lennard
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hardy Spaces ◽  
Weak Topology ◽  
Uniformly Convex ◽  
Vector Valued

AbstractIn [8] Partington showed that a Banach space X is uniformly convex if and only if Lp([0, 1], X) has the uniform Kadec–Klee–Huff property with respect to the weak topology (UKKH (weak)), where 1 < p < ∞. In this note we will characterize the Banach spaces X such that HP(D, X) has UKKH (weak), where 1 ≤ p < ∞. Similar results for UKKH (weak*) are also obtained. These results (and proofs) are quite different from Partington's result (and proof).

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