# Questions tagged [banach-lattices]

Banach lattices are Banach spaces endowed with a partial ordering that is compatible with the norm.

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### Closed subspaces of $L^1(\mathbb{R})$ that is not isomorphic to a subspace of a space with an unconditional basis

It is known that $L^1([0,1])$ does not admit any unconditional basis. Even more, $L^1([0,1])$ is not isomorphic to a subspace of a space with an unconditional basis. My question is the following ...
1 vote
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### Positivity of the bidual operator

How is the fixed space of a bounded linear operator related to the fixed space of the bidual operator? Motivation of question: I have a strongly continuous semigroup $(T(t))_{t\ge 0}$ on a Banach ...
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### Kernel operators on AL-spaces

Let $E$ be an AL-space. For simplicity $E=L_{1}(X,\Sigma,\mu)$, where $(X,\Sigma,\mu)$ is a strictly localizable measure space. Let $T:E\rightarrow E$ be a bounded kernel operator on $E$ with ...
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### Is every AM space a $C^*$-algebra?

A Banach lattice $E$ is said to be an AM-space if $$\|\sup\{x,y\}\|=\sup\{\|x\|,\|y\|\}$$ for all positive $x,y\in E$. My question is as follows: Is every AM-space (which is a $*$-algebra) a $C^*$-...
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### Eigenvectors of the dual of positive irreducible operators

Let $E$ be a Banach Lattice such that $E$ is a $M$-space. Assume that $T:E\to E$ is a positive bounded non-compact irreducible linear operator with positive spectral radius. And define \begin{align*} ...
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### Pre-dual of a lattice is a lattice?

Let $X$ be a Banach space such that its dual $X'$ is a Banach lattice. Then must $X$ also be a lattice? I know that if $X$ is a lattice, then so is its dual $X'$. However, I was wondering is the ...
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### Banach lattice from a positive linear form on the dual

I'm trying to understand the following construction: Let $E$ a Banach lattice. Basically we want to construct for each $\nu \in E^*$ (A positive linear functional $\nu:E \rightarrow \mathbb{C})$, a ...
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### Examples of Normalized Semigroup of Operators

In the theory of semigroups of bounded linear operators ${Z(t)}_{t\geq0}$, the normalized semigroups, $Z(t) : V\rightarrow V$ are defined as follows: $$Z(t)e=e,$$ where $V$ is unital Banach Algebra ...
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### Positiveness of Inverse Of Positive Operator Implies Lattice Isomorphism?

Let $X$ be a Banach Lattice and denote by $\mathcal{B}(X)$ the banach space of bounded linear endomorphisms. An operator $T \in \mathcal{B}(X)$ is called positive if $Tx \geq 0$ whenever $x \geq 0.$ ...
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### Product of two weakly compact endomorphisms is compact

I've seen the statement that if a Banach lattice $E$ satisfies the property that $$||x + y|| = ||x|| + ||y||, \: \forall x,y \in E_+$$ then if $S,T:E \to E$ are weakly compact endomorphisms, then $ST$...
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### Weakly null sequence in Banach lattices

Let $(x_n)_n$ be a positive, disjoint, weakly null sequence in a Banach lattice $E$. If $(y_n)_n$ is a sequence such that $0\leq y_n\leq x_n$ for every $n\in \mathbb{N}$, we can garantee that $y_n$ is ...
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### closed lattice ideal is isomorphic to $C(K)$

Let $X\in E$, where $E$ is a Banach function space on $(0,1)$. Consider the interval $[-X,X]$ and generate it to a closed lattice ideal $I$ of $E$. We may renorm this ideal $I$ such that $[-X,X]$ is ...
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### Establish Archimedean property of a vector-lattice

I am trying to find ways to prove the Archimedian property of a certain vector lattice and got stuck on the following type of problem. I feel the statement below (or in fact weaker versions) should ...
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### Unbounded norm convergence

I study on unbounded norm convergence in Banach lattice. I have to find an example.Is there any example of unbounded norm convergence but not norm convergence in Banach lattice?
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### Reference for Banach Lattices

I am looking for a survey or a basic book on Banach Lattices (I know next to nothing about the subject, so a basic survey is what I believe will be helpful). Can somebody please suggest any good ...
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### Infinite dimensional Banach lattice $L^\infty(X)$ is not order continuous

Consider an arbitrary measure space $(X,\Sigma,\mu)$, with the only assumption being that $L^\infty(X)$ is infinite dimensional. Consider $L^\infty(X)$ as a Banach lattice with the usual ordering. As ...
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### Hardy space as a Banach lattice

The Hardy spaces $H^p$ of holomorphic functions on the unit disk are Banach spaces. Question: Are they also Banach lattices? If yes, why is it less common to consider the Hardy spaces as Banach ...
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### Show that two projections commute if and only if their respectively closed subspaces are compatibles

I have the next problem: Let $\mathcal{H}$ be a separable and complex Hilbert space, with $S$ and $Q$ two closed subspaces on it. Let $P_S$ and $P_Q$ be the orthogonal projection onto $S$ and $Q$ ...
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### Strong convergence of regular operators and convergence of the modulus

Let $E$ be a Banach lattice and $T_n\in\mathcal{L}^r(E)$ a sequence of regular operators such that $T_n$ converges strongly to $T\in\mathcal{L}^r(E)$. How to prove that $\left|T_n\right|$ converges ...
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### "Representation Capacity" of Finite Lattice Ordered Modules

Apologies for what is probably a very basic question, but I am looking for references for what sort of lattices can be represented in what I think should be called something like "finite lattice-...
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### Positive cone of Banach lattice algebra

From the literature on Banach lattice algebras (and that on ordered Banach algebras) there does not appear to be a consensus on a definition. What is agreed is that one should be a Banach lattice, be ...
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### Division in Banach Lattice Algebra

Let X be a Banach Lattice Algebra and $X_+=\{f\in X: f>0\}$. Let $f:X_+\rightarrow X$ be continuously differentiable. Question: When does the expression $\frac{f'(x)}{x}$ for $x\in X_+$, make ...
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### A question about passing to limits in Banach lattices

I would be grateful if one could confirm that the following argumentation is fine. Suppose that $L$ is a Banach lattice and $(L_n)$ is an increasing sequence of sublattices of $L$. Given two positive ...
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### Are dual spaces with unconditional bases weakly sequentially complete?

It is well known that a weakly sequentially complete Banach space with an unconditional basis is isomorphic to a conjugate space. Is the converse to this statement true? If a Banach space is a ...
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