# Banach Lattice Structures and Concavifications in Banach Spaces

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## Abstract

**:**

## 1. Introduction

## 2. Standard Definitions and Basic Concepts

#### 2.1. Banach Function Spaces, pth Powers, and Vector Measures

#### 2.2. The Basic Construction

**Definition**

**1.**

**Remark**

**1.**

**Lemma**

**1.**

**Proof.**

**Example**

**1.**

- (i)
- Consider a pair of Banach function spaces $X(\mu )$ and $Y(\mu )$ over a finite measure space $(\mathsf{\Omega},\mathsf{\Sigma},\mu )$. Take the space$$M(X(\mu ),Y(\mu ))=\left\{g\in {L}^{0}(\mu ):fg\in Y(\mu ),\phantom{\rule{0.166667em}{0ex}}f\in X(\mu )\right\}$$
- (ii)
- Consider a finite measure space $(\mathsf{\Omega},\mathsf{\Sigma},\mu )$ and a Banach space F. Take an order-to-norm continuous operator $T:{L}^{\infty}(\mu )\to F$ and consider the canonical vector measure ${m}_{T}:\mathsf{\Sigma}\to F$ given by ${m}_{T}(A)=T({\chi}_{A})$, $A\in \mathsf{\Sigma}$. Assume also that T is μ-determined, i.e., $\parallel {m}_{T}\parallel (A)=0$ if and only if $\mu (A)=0$. Using the information about the computation of the norm in ${L}^{1}({m}_{T})$ given in Section 2.1, we find that$${I}_{{m}_{T}}\circ {M}_{g}(f):={I}_{{m}_{T}}(gf)$$$$\parallel {I}_{{m}_{T}}\circ {M}_{g}{\parallel}_{L({L}^{\infty}(\mu ),F)}={\parallel g\parallel}_{{L}^{1}({m}_{T})}.$$$$E:=\left\{{I}_{{m}_{T}}\circ {M}_{g}\in L({L}^{\infty}(\mu ),F):\phantom{\rule{0.166667em}{0ex}}g\in {L}^{1}({m}_{T})\right\},$$

**Definition**

**2.**

**Example**

**2.**

## 3. Order-Continuous Banach Function Subspaces and $\mathit{p}$th Powers of Banach Spaces

**Theorem**

**1.**

- (1)
- There is a finite measure μ and an order-continuous Banach function space $Y(\mu )$ such that E is representable on $Y(\mu )$.
- (2)
- There is a measurable space $(\mathsf{\Omega},\mathsf{\Sigma})$ and a vector measure $m:\mathsf{\Sigma}\to E$ such that
- (i)
- $span\{\mathrm{rg}(m)\}$ is dense in E, and
- (ii)
- for every sequence $({f}_{n})$ of simple functions, if ${({I}_{m}({f}_{n}))}_{n}$ is Cauchy in E, then ${({f}_{n})}_{n}$ is a Cauchy sequence in ${L}^{1}(m)$ too.

**Proof.**

**Example**

**3.**

**Proposition**

**1.**

- (1)
- $\overline{\iota (Y(\mu ))}$ is a Banach subspace of E that can be represented over a Banach function space $Z(\mu )$ containing $Y(\mu )$.
- (2)
- There is a constant $K>0$ such that for every simple function $f\in Y(\mu )$,$$\underset{h\in {B}_{{L}^{\infty}(\mu )}}{sup}{\parallel \iota (fh)\parallel}_{E}\le K{\parallel \iota (f)\parallel}_{E}.$$

**Proof.**

**Example**

**4.**

**Definition**

**3.**

**Proposition**

**2.**

**Proof.**

**Corollary**

**1.**

- (i)
- The function ${\parallel \xb7\parallel}_{{E}_{\left[p\right]}}$ is in fact a norm, and
- (ii)
- $({E}_{\left[p\right]}{,\parallel \xb7\parallel}_{{E}_{\left[p\right]}}{)=(Y(\mu )}_{\left[p\right]}{,\parallel \xb7\parallel}_{Y{(\mu )}_{\left[p\right]}})$ isometrically.

**Proof.**

**Corollary**

**2.**

**Proof.**

**Corollary**

**3.**

**Corollary**

**4.**

**Proof.**

**Corollary**

**5.**

**Proof.**

**Corollary**

**6.**

**Proof.**

**Corollary**

**7.**

**Proof.**

## 4. Applications: Banach Lattice Structures in Banach Spaces

#### 4.1. p-Concave Order-Continuous Banach Spaces

**Corollary**

**8.**

- (1)
- There is a finite measure μ and an order-continuous p-concave Banach function space $Y(\mu )$ such that E is representable on $Y(\mu )$.
- (2)
- There is a measurable space $(\mathsf{\Omega},\mathsf{\Sigma})$, a constant $K>0$, and a vector measure $m:\mathsf{\Sigma}\to E$ such that
- (i)
- for every simple function f and $A\in \mathsf{\Sigma}$, $\parallel {I}_{m}(f{\chi}_{A})\parallel \le \parallel {I}_{m}(f)\parallel $, and
- (ii)
- for every finite set ${f}_{1},\dots ,{f}_{N}$ of simple functions,$${(\sum _{n=1}^{N}{\parallel {f}_{n}\parallel}_{{L}^{1}(m)}^{p})}^{1/p}\le K\parallel {I}_{m}((\sum _{n=1}^{N}|{f}_{n}{{|}^{p})}^{1/p}){\parallel}_{E}.$$

**Proof.**

**Corollary**

**9.**

**Proof.**

**Corollary**

**10.**

**Proof.**

#### 4.2. Banach Function Subspaces of Multiplication Operators between Banach Spaces

**Theorem**

**2.**

- (1)
- There is a subspace E of $L(X(\mu ),F)$ that is isomorphic to $Y(\mu )$, and the isomorphism $\iota :Y(\eta )\to E$ satisfies that $\iota ({\chi}_{A})(\xb7)=\iota ({\chi}_{\mathsf{\Omega}})({\chi}_{A}\phantom{\rule{0.166667em}{0ex}}(\xb7))$.
- (2)
- There is a μ-determined operator $T\in L(X(\mu ),F)$ such that$$Y(\mu )={M}_{0}(X(\mu ),{L}^{1}({m}_{T})),$$

**Proof.**

#### 4.3. Orthogonal Polynomials on Banach Spaces

**Corollary**

**11.**

**Proof.**

**Corollary**

**12.**

**Proof.**

**Corollary**

**13.**

**Proof.**

**Corollary**

**14.**

**Proof.**

**Remark**

**2.**

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Agud, L.; Calabuig, J.M.; Juan, M.A.; Sánchez Pérez, E.A.
Banach Lattice Structures and Concavifications in Banach Spaces. *Mathematics* **2020**, *8*, 127.
https://doi.org/10.3390/math8010127

**AMA Style**

Agud L, Calabuig JM, Juan MA, Sánchez Pérez EA.
Banach Lattice Structures and Concavifications in Banach Spaces. *Mathematics*. 2020; 8(1):127.
https://doi.org/10.3390/math8010127

**Chicago/Turabian Style**

Agud, Lucia, Jose Manuel Calabuig, Maria Aranzazu Juan, and Enrique A. Sánchez Pérez.
2020. "Banach Lattice Structures and Concavifications in Banach Spaces" *Mathematics* 8, no. 1: 127.
https://doi.org/10.3390/math8010127