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.

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 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 Classes of operators on vector valued integration spaces

1977 ◽  
Vol 24 (2) ◽  
pp. 129-138 ◽  
Author(s):  
R. J. Fleming ◽  
J. E. Jamison
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Measure Space ◽  
Separable Hilbert Space ◽  
Hermitian Operator ◽  
Finite Measure ◽  
Finite Measure Space ◽  
Image Position ◽  
Vector Valued ◽  
Uniformly Bounded

AbstractLet Lp(Ω, K) denote the Banach space of weakly measurable functions F defined on a finite measure space and taking values in a separable Hilbert space K for which ∥ F ∥p = ( ∫ | F(ω) |p)1/p < + ∞. The bounded Hermitian operators on Lp(Ω, K) (in the sense of Lumer) are shown to be of the form , where B(ω) is a uniformly bounded Hermitian operator valued function on K. This extends the result known for classical Lp spaces. Further, this characterization is utilized to obtain a new proof of Cambern's theorem describing the surjective isometries of Lp(Ω, K). In addition, it is shown that every adjoint abelian operator on Lp(Ω, K) is scalar.

scholarly journals Essential supremum norm differentiability

1985 ◽  
Vol 8 (3) ◽  
pp. 433-439
Author(s):  
I. E. Leonard ◽  
K. F. Taylor
Keyword(s):  
Banach Space ◽  
Measure Space ◽  
Real Banach Space ◽  
Finite Measure ◽  
Linear Operators ◽  
Supremum Norm ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Fréchet Differentiability ◽  
Finite Measure Space

The points of Gateaux and Fréchet differentiability inL∞(μ,X)are obtained, where(Ω,∑,μ)is a finite measure space andXis a real Banach space. An application of these results is given to the spaceB(L1(μ,ℝ),X)of all bounded linear operators fromL1(μ,ℝ)intoX.

scholarly journals Best N-Simultaneous Approximation in Lp(μ,X)

2017 ◽  
Vol 2017 ◽  
pp. 1-4
Author(s):  
Tijani Pakhrou
Keyword(s):  
Banach Space ◽  
Compact Subset ◽  
Measure Space ◽  
Simultaneous Approximation ◽  
Finite Measure ◽  
Weakly Compact ◽  
Integrable Functions ◽  
Finite Measure Space

Let X be a Banach space. Let 1≤p<∞ and denote by Lp(μ,X) the Banach space of all X-valued Bochner p-integrable functions on a certain positive complete σ-finite measure space (Ω,Σ,μ), endowed with the usual p-norm. In this paper, the theory of lifting is used to prove that, for any weakly compact subset W of X, the set Lp(μ,W) is N-simultaneously proximinal in Lp(μ,X) for any arbitrary monotonous norm N in Rn.

scholarly journals Weighted p-regular kernels for reproducing kernel Hilbert spaces and Mercer Theorem

2019 ◽  
Vol 18 (03) ◽  
pp. 359-383
Author(s):  
L. Agud ◽  
J. M. Calabuig ◽  
E. A. Sánchez Pérez
Keyword(s):  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Banach Function Space ◽  
Finite Measure ◽  
Reproducing Kernel Hilbert Spaces ◽  
Kernel Operator ◽  
Finite Measure Space ◽  
Kernel Hilbert Spaces ◽  
Vector Valued Function ◽  
Weighted Kernels

Let [Formula: see text] be a finite measure space and consider a Banach function space [Formula: see text]. Motivated by some previous papers and current applications, we provide a general framework for representing reproducing kernel Hilbert spaces as subsets of Köthe–Bochner (vector-valued) function spaces. We analyze operator-valued kernels [Formula: see text] that define integration maps [Formula: see text] between Köthe–Bochner spaces of Hilbert-valued functions [Formula: see text] We show a reduction procedure which allows to find a factorization of the corresponding kernel operator through weighted Bochner spaces [Formula: see text] and [Formula: see text] — where [Formula: see text] — under the assumption of [Formula: see text]-concavity of [Formula: see text] Equivalently, a new kernel obtained by multiplying [Formula: see text] by scalar functions can be given in such a way that the kernel operator is defined from [Formula: see text] to [Formula: see text] in a natural way. As an application, we prove a new version of Mercer Theorem for matrix-valued weighted kernels.

scholarly journals Best approximation in Orlicz spaces

1991 ◽  
Vol 14 (2) ◽  
pp. 245-252 ◽  
Author(s):  
H. Al-Minawi ◽  
S. Ayesh
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Best Approximation ◽  
Measure Space ◽  
Real Banach Space ◽  
Closed Subspace ◽  
Orlicz Spaces ◽  
Finite Measure ◽  
Reflexive Subspace ◽  
Finite Measure Space

LetXbe a real Banach space and(Ω,μ)be a finite measure space andϕbe a strictly icreasing convex continuous function on[0,∞)withϕ(0)=0. The spaceLϕ(μ,X)is the set of all measurable functionsfwith values inXsuch that∫Ωϕ(‖c−1f(t)‖)dμ(t)<∞for somec>0. One of the main results of this paper is: “For a closed subspaceYofX,Lϕ(μ,Y)is proximinal inLϕ(μ,X)if and only ifL1(μ,Y)is proximinal inL1(μ,X)′​′. As a result ifYis reflexive subspace ofX, thenLϕ(ϕ,Y)is proximinal inLϕ(μ,X). Other results on proximinality of subspaces ofLϕ(μ,X)are proved.

Complemented copies of l1, in Lp(μ;E)

1992 ◽  
Vol 111 (3) ◽  
pp. 531-534 ◽  
Author(s):  
José Mendoza
Keyword(s):  
Banach Space ◽  
Measure Space ◽  
Finite Measure ◽  
Complemented Subspace ◽  
Integrable Functions ◽  
Usual Norm ◽  
Finite Measure Space

AbstractLet E be a Banach space, let (Ω, Σ, μ) a finite measure space, let 1 < p < ∞ and let Lp(μ;E) the Banach space of all E-valued p-Bochner μ-integrable functions with its usual norm. In this note it is shown that E contains a complemented subspace isomorphic to l1 if (and only if) Lp(μ; E) does. An extension of this result is also given.

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