A new class of restricted type spaces

2011 ◽  
Vol 54 (3) ◽  
pp. 749-759
Author(s):  
Salvador Rodríguez-López ◽  
Javier Soria
Keyword(s):  
Banach Space ◽  
Function Space ◽  
Lorentz Space ◽  
Banach Function Space ◽  
Hardy Operator ◽  
Best Constant ◽  
New Class ◽  
Type Spaces ◽  
Restricted Type

AbstractWe find new properties for the space R(X), introduced by Soria in the study of the best constant for the Hardy operator minus the identity. In particular, we characterize when R(X) coincides with the minimal Lorentz space Λ(X). The condition that R(X) ≠ {0} is also described in terms of the embedding (L1, ∞ ∩ L∞) ⊂ X. Finally, we also show the existence of a minimal rearrangement-invariant Banach function space (RIBFS) X among those for which R(X) ≠ {0} (which is the RIBFS envelope of the quasi-Banach space L1, ∞ ∩ L∞).

A Stability Property of a Class of Banach Spaces Not Containing c0

1992 ◽  
Vol 35 (1) ◽  
pp. 56-60 ◽  
Author(s):  
Patrick N. Dowling
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Function Space ◽  
Stability Property ◽  
Banach Function Space ◽  
Ideal Space

AbstractLet E be a Banach ideal space and X be a Banach space. The Banach function space E(X) does not contain a copy of C0 if and only if neither E nor X contains a copy of c0. Some extensions of this result are also noted.

scholarly journals Banach Lattice Structures and Concavifications in Banach Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 127
Author(s):  
Lucia Agud ◽  
Jose Manuel Calabuig ◽  
Maria Aranzazu Juan ◽  
Enrique A. Sánchez Pérez
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Function Space ◽  
Lattice Structure ◽  
Banach Function Space ◽  
Finite Measure ◽  
Multiplication Operators ◽  
Spaces Of Operators ◽  
Finite Measure Space ◽  
Local Properties

Let ( Ω , Σ , μ ) be a finite measure space and consider a Banach function space Y ( μ ) . We say that a Banach space E is representable by Y ( μ ) if there is a continuous bijection I : Y ( μ ) → E . In this case, it is possible to define an order and, consequently, a lattice structure for E in such a way that we can identify it as a Banach function space, at least regarding some local properties. General and concrete applications are shown, including the study of the notion of the pth power of a Banach space, the characterization of spaces of operators that are isomorphic to Banach lattices of multiplication operators, and the representation of certain spaces of homogeneous polynomials on Banach spaces as operators acting in function spaces.

scholarly journals Banach spaces with property (w)

1993 ◽  
Vol 35 (2) ◽  
pp. 207-217 ◽  
Author(s):  
Denny H. Leung
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Banach Lattice ◽  
Banach Lattices ◽  
Weakly Compact ◽  
Type Spaces

A Banach space E is said to have Property (w) if every operator from E into E' is weakly compact. This property was introduced by E. and P. Saab in [9]. They observe that for Banach lattices, Property (w) is equivalent to Property (V*), which in turn is equivalent to the Banach lattice having a weakly sequentially complete dual. Thus the following question was raised in [9].Does every Banach space with Property (w) have a weakly sequentially complete dual, or even Property (V*)?In this paper, we give two examples, both of which answer the question in the negative. Both examples are James type spaces considered in [1]. They both possess properties stronger than Property (w). The first example has the property that every operator from the space into the dual is compact. In the second example, both the space and its dual have Property (w). In the last section we establish some partial results concerning the problem (also raised in [9]) of whether (w) passes from a Banach space E to C(K, E).

scholarly journals Almost Surjective Epsilon-Isometry in The Reflexive Banach Spaces

CAUCHY ◽  
2017 ◽  
Vol 4 (4) ◽  
pp. 167
Author(s):  
Minanur Rohman
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Lorentz Space ◽  
Bounded Linear Operator ◽  
Closed Subspace ◽  
Reflexive Banach Space ◽  
Reflexive Banach Spaces ◽  
Bounded Linear ◽  
Star Topology

<p class="AbstractCxSpFirst">In this paper, we will discuss some applications of almost surjective epsilon-isometry mapping, one of them is in Lorentz space ( L_(p,q)-space). Furthermore, using some classical theorems of w star-topology and concept of closed subspace -complemented, for every almost surjective epsilon-isometry mapping  <em>f </em>: <em>X to</em><em> Y</em>, where <em>Y</em> is a reflexive Banach space, then there exists a bounded linear operator   <em>T</em> : <em>Y to</em><em> X</em>  with  such that</p><p class="AbstractCxSpMiddle">  </p><p class="AbstractCxSpLast">for every x in X.</p>

scholarly journals Commutators of the Bilinear Hardy Operator on Herz Type Spaces with Variable Exponents

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Shengrong Wang ◽  
Jingshi Xu
Keyword(s):  
Morrey Spaces ◽  
Hardy Operator ◽  
Variable Exponents ◽  
Herz Spaces ◽  
Bmo Functions ◽  
Type Spaces

In this paper, we obtain the boundedness of bilinear commutators generated by the bilinear Hardy operator and BMO functions on products of Herz spaces and Herz-Morrey spaces with variable exponents.

scholarly journals On the uniform Kadec-Klee property with respect to convergence in measure

1995 ◽  
Vol 59 (3) ◽  
pp. 343-352 ◽  
Author(s):  
F. A. Sukochev
Keyword(s):  
Function Space ◽  
Symmetric Function ◽  
Lorentz Space ◽  
Symmetric Operator ◽  
Von Neumann Algebra ◽  
Finite Measure ◽  
Operator Space ◽  
Von Neumann ◽  
Convergence In Measure ◽  
Uniform Monotonicity

AbstractLet E(0, ∞) be a separable symmetric function space, let M be a semifinite von Neumann algebra with normal faithful semifinite trace μ, and let E(M, μ) be the symmetric operator space associated with E(0, ∞). If E(0, ∞) has the uniform Kadec-Klee property with respect to convergence in measure then E(M, μ) also has this property. In particular, if LΦ(0, ∞) (ϕ(0, ∞)) is a separable Orlicz (Lorentz) space then LΦ(M, μ) (Λϕ (M, μ)) has the uniform Kadec-Klee property with respect to convergence in measure on sets of finite measure if and only if the norm of E(0, ∞) satisfies G. Birkhoff's condition of uniform monotonicity.

scholarly journals Nuclear operators on Banach function spaces

Positivity ◽  
2020 ◽  
Author(s):  
Marian Nowak
Keyword(s):  
Banach Space ◽  
Convex Space ◽  
Banach Function Space ◽  
Finite Measure ◽  
Locally Convex Space ◽  
Nuclear Operator ◽  
Representing Measure ◽  
Nuclear Operators ◽  
Nikodym Property ◽  
Locally Convex

Abstract Let X be a Banach space and E be a perfect Banach function space over a finite measure space $$(\Omega ,\Sigma ,\lambda )$$ ( Ω , Σ , λ ) such that $$L^\infty \subset E\subset L^1$$ L ∞ ⊂ E ⊂ L 1 . Let $$E'$$ E ′ denote the Köthe dual of E and $$\tau (E,E')$$ τ ( E , E ′ ) stand for the natural Mackey topology on E. It is shown that every nuclear operator $$T:E\rightarrow X$$ T : E → X between the locally convex space $$(E,\tau (E,E'))$$ ( E , τ ( E , E ′ ) ) and a Banach space X is Bochner representable. In particular, we obtain that a linear operator $$T:L^\infty \rightarrow X$$ T : L ∞ → X between the locally convex space $$(L^\infty ,\tau (L^\infty ,L^1))$$ ( L ∞ , τ ( L ∞ , L 1 ) ) and a Banach space X is nuclear if and only if its representing measure $$m_T:\Sigma \rightarrow X$$ m T : Σ → X has the Radon-Nikodym property and $$|m_T|(\Omega )=\Vert T\Vert _{nuc}$$ | m T | ( Ω ) = ‖ T ‖ nuc (= the nuclear norm of T). As an application, it is shown that some natural kernel operators on $$L^\infty $$ L ∞ are nuclear. Moreover, it is shown that every nuclear operator $$T:L^\infty \rightarrow X$$ T : L ∞ → X admits a factorization through some Orlicz space $$L^\varphi $$ L φ , that is, $$T=S\circ i_\infty $$ T = S ∘ i ∞ , where $$S:L^\varphi \rightarrow X$$ S : L φ → X is a Bochner representable and compact operator and $$i_\infty :L^\infty \rightarrow L^\varphi $$ i ∞ : L ∞ → L φ is the inclusion map.

scholarly journals The weak cotype 2 and the Orlicz property of the Lorentz sequence space d(a, 1)

1992 ◽  
Vol 34 (3) ◽  
pp. 271-276
Author(s):  
J. Zhu
Keyword(s):  
Banach Space ◽  
Function Space ◽  
Sequence Space ◽  
Affirmative Answer ◽  
Sequence Spaces ◽  
Similar Question ◽  
Lorentz Sequence Space ◽  
Symmetric Basis ◽  
Orlicz Property

The question “Does a Banach space with a symmetric basis and weak cotype 2 (or Orlicz) property have cotype 2?” is being seriously considered but is still open though the similar question for the r.i. function space on [0, 1] has an affirmative answer. (If X is a r.i. function space on [0, 1] and has weak cotype 2 (or Orlicz) property then it must have cotype 2.) In this note we prove that for Lorentz sequence spaces d(a, 1) they both hold.

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