Questions tagged [banach-lattices]

A Banach lattice is a complete normed vector lattice such that the ordering and norm are compatible.

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6
votes
1answer
120 views

Are lattice operations in a Lipschitz space sequentially continuous in the weak* topology?

This is a follow-up on this (answered) question on math.SE, but involves a different topology. I think this time it is more appropriate for MO. I will repeat the background from the question cited ...
4
votes
1answer
137 views

Consider a net of weak order units in a Riesz space converging in order to a weak order unit. Is there a tail whose infimum is a weak order unit?

Let $X$ be an extremally disconnected (the closure of an open set is open) compact Hausdorff space, and consider the Riesz space $C^\infty(X)$ of continuous functions from $X$ to the extended real ...
2
votes
2answers
94 views

lattice suprema vs pointwise suprema

What is the difference between the lattice supremum and the pointwise supremum of a family of functions? I mean, given a family of real valued functions $\mathcal{F}$, is the function $\sup\mathcal{F}:...
2
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0answers
88 views

Is a closed connected semilattice of $C(I)$ path-connected?

Let $\Gamma $ be a sub-lattice of the Banach space $\big( B(S),\|\cdot\|_\infty\big)$ of all bounded real valued functions on the set $S$ (meaning that for any $f,g\in\Gamma $ both functions $f\wedge ...
2
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0answers
64 views

Principal ideals in Banach lattices

Let $E$ be a Banach lattice. Then $u \in E_+$ is said to be a quasi-interior point of $E$ is $$E_u:=\{f \in E:\exists\, c\geq 0 \text{ such that } |f| \leq cu\}$$ is dense in $E.$ Let $\Omega$ be a ...
5
votes
0answers
57 views

Reference request: a survey of (linear) Krein-Rutman theory

I'm looking for a survey article or book chapter where a rather exhaustive treatment of the Krein-Rutman theory of positive linear operators an ordered Banach spaces is given. Motivation. Some ...
2
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0answers
65 views

Banach lattices $X$ for which $L_p(\mu)\subset X$ or $X\subset L_q(\mu)$

It is well known (see vol. II of Lindenstrauss and Tzafriri's book) that an order continuous Banach lattice $X$ with a weak unit admits a representation as a (in general not closed) ideal of $L_1(\mu)$...
3
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1answer
250 views

Copies of $c_0$ in $C[0,1]$ and disjoint sequences

Let $M$ be a subspace of $C[0,1]$ isomorphic to $c_0$. QUESTION: Is it possible to find a normalized disjoint sequence $(f_n)$ in $C[0,1]$ such that the distance of $f_n$ to $M$ tends to $0$ as $n$ ...
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0answers
40 views

Is there a multiplier rule for this minimization problem?

Let $(E,\mathcal E)$ be a measurable space, $W\subseteq\left\{w:E\to\mathbb R\mid w\text{ is }\mathcal E\text{-measurable}\right\}$ be a Banach space, $k\in\mathbb N$ and $f:W^k\to[0,\infty)$. I'm ...
6
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0answers
85 views

$\ell^\infty / ces_0$ as an ordered Banach space

Let $ces_0:=\{\xi\in\ell^\infty : \lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^{n}\xi_k=0\}$ and $q:\ell^\infty \to \ell^\infty/ces_0$ be the usual quotient map. The space $ces_0$ is closed in $\ell^\...
5
votes
1answer
182 views

Complemented subspaces of Lorentz sequence spaces?

Let $d(\textbf{w},p)$, $1\leq p<\infty$, denote the Lorentz sequence space, where $\textbf{w}=(w_n)_{n=1}^\infty\in c_0\setminus\ell_1$ is a normalized decreasing weight. Is there very much known ...
3
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0answers
26 views

Borel rank of certain automorphism orbits in $L_p$ lattices

For any $p$ with $1\leq p < \infty$, let $L_p([0,1])$ be the Banach lattice of $L_p$ functions on the unit interval (with the standard measure). Let $A=\{f\in L_p([0,1]):\left\lVert f \right \...
3
votes
1answer
150 views

Reference request: Spectral properties of real operators

Let $A:D(A)\subseteq E \to E$ be a closed operator on a complex Banach lattice $E.$ Then $A$ is said to be real if $x+iy \in D(A) \implies x,y \in D(A)$ for all $x,y \in E_{\mathbb R}$ and $A(D(A) \...
3
votes
1answer
235 views

Reference request: Irreducible operators

I had asked this question on MSE but did not get any response. I would like some reference to books that talk about irreducible operators on Banach lattices and its properties. A quick google search ...
3
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0answers
108 views

Classical subspaces of non-atomic Banach lattices

Tsirelson's space was the first example of a Banach space which does not have a subspace isomoprhic to any of the classical spaces $\ell_p$, $1\leqslant p<\infty$, or $c_0$. As this space has a $...
4
votes
1answer
272 views

Weak compactness of order intervals in $L^1$

Let $(\Omega,\mu)$ be a measure space, say $\sigma$-finite for the sake of simplicity, and let $L^1 := L^1(\Omega,\mu)$ denote the real-valued $L^1$-space over $(\Omega,\mu)$. For all $f,h \in L^1$ ...
5
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0answers
157 views

Examples of Banach lattices with positive Schur property but without Schur property

A Banach lattice $E$ has the $(1)$ Schur property provided that any weakly null sequence in $E$ is norm null in $E$, and $(2)$ positive Schur property provided that any weakly null sequence of ...
0
votes
1answer
96 views

Definition of an Orlicz modular space

In Nowak (1989), a modular $\rho$ on a vector lattice is defined by the following properties (N1) $\rho(x)=0\implies x=0$; (N2) $\lvert x\rvert \le \lvert y\rvert\implies \rho(x) \le \rho(y)$; (N3) ...
2
votes
0answers
73 views

Linearly Isometric Banach lattices

Do there exist two Banach lattices which are linearly isometric but not Banach-lattice isometric? This is so basic that you would expect a textbook or monograph to have answered this, but I have not ...
2
votes
0answers
115 views

Semigroups on Banach lattice

Let $\{Z(t)\}_{t\geq 0}$ be a semigroup of positive operators on a Banach lattice $X$. I want to show that $$\int_0^1[Z(1-s)+Z(1+s)]fds-f\in X_+,\quad f\in X_+$$ Where $X_+$ denotes the positive ...
2
votes
1answer
232 views

Positive semigroups and convex operator

Let $\{Z(t)\}_{t\geq 0}$ be a strongly continuous positive semigroup on a Banach lattice $V$ (endowed with ordering $\leq$). Let $\phi:V\rightarrow V$ be a convex operator. I want to prove that $$\phi(...