# Questions tagged [banach-lattices]

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

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

36
questions

6
votes

2
answers

196
views

Let $(X, \mu)$ be your favourite measure space (finite or $\sigma$-finite if you like), let $g \in L^2$ (say, the scalar field of $L^2$ is $\mathbb{R}$, though this probably doesn't matter). Let $\...

2
votes

0
answers

46
views

Let $E$ be an AL space (i.e. a Banach lattice whose norm is additive on the positive cone $E_+$) that satisfies Mazur's condition (every sequentially weak$^*$-continuous functional on $E'$ is weak$^*$-...

1
vote

1
answer

51
views

Let $W$ be a positive non-increasing continuous function on $(0,1]$ so
that $\lim_{t \rightarrow 0} W(t)=\infty$, $W(1)=1$ and $\int_0^1 W(t) dt =1$.
For $1 \leq p <\infty$, the Lorentz function ...

4
votes

0
answers

56
views

Given a Banach space $X$ and a non-trivial ultrafilter $\mathcal{U}$ on a set $I$, the ultrapower $X_\mathcal{U}$ is defined as the quotient of $\ell_\infty(I,X)$ by the closed subspace $N_\mathcal{U}(...

4
votes

1
answer

138
views

Let $\mu$ be a finite measure, like the Lebesgue measure in $(0,1)$. It is well-known that $L_1(\mu)$ and its second dual $L_1(\mu)^{**}$ are Banach lattices, $L_1(\mu)$ is a projection band in $L_1(\...

2
votes

0
answers

45
views

Let $K$ be a compact Hausdorff space and let $U,V\subset K$ be open. Let $\left(f_{i}\right)_{i\in I}$ be an increasing net of continuous non-negative functions such that $f_{i}\le 1$ and $f_{i}$ ...

2
votes

0
answers

57
views

It is well-known that any Banach rearrangement-invariant function space $X$ on $[0,1]$ is a subset of $L_1[0,1]$, and I can find a reference that any quasi-Banach rearrangement-invariant function ...

1
vote

1
answer

78
views

This question was previously posted on MSE.
Let $E$ be a Banach lattice such that $E$ is an $M$-space. Assume that $T\colon E\to E$ is a positive bounded non-compact irreducible linear operator with ...

2
votes

1
answer

139
views

Let $X$ and $Y$ be compact Hausdorff spaces and let $\varphi:X\to Y$ be continuous with a property that if $A$ is a nowhere dense zero-set in $Y$, then $\varphi^{-1}(A)$ is nowhere dense in $X$. Let $...

3
votes

1
answer

117
views

I looked in all textbooks on vector lattices (Riesz spaces) as well as ordered vector spaces, but couldn't find any mentions of neither inductive nor projective limit for these structures. Googling ...

3
votes

1
answer

165
views

I am looking for a paper "Linear spaces with disjoint elements and their conversion into vector lattices" by A. I. Veksler.
It was published in 1967 in Research Notes of Leningrad State ...

2
votes

1
answer

98
views

Let $(\Omega,\Sigma,\mu)$ a finite measure space. We want to characterize the order convergence (for sequences) in Bochner integrable functions space $L^1(\mu,X)$, $X$ Banach lattice.
In $L^p$ we have:...

2
votes

1
answer

202
views

Let $X$, $Y_1$ and $Y_2$ be a compact Hausdorff spaces and let $\varphi_i:X\to Y_i$ be a continuous surjection (and so a quotient map).
Let $\sim$ be the minimal closed equivalence relation on $X$ ...

4
votes

1
answer

383
views

Let $B$ be a complex Banach lattice and let $T : B \to B$ be a positive operator. Are there any conditions that ensure that $T$ has an eigenvalue? I am interested in particular in non-compact ...

9
votes

1
answer

225
views

The title says it all:
Let $E$ be a Banach lattice, which is isomorphic to a Hilbert space (as normed spaces). Is there an equivalent Hilbert norm on $E$, which still makes it a Banach lattice with ...

6
votes

1
answer

172
views

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

1
answer

155
views

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

2
answers

294
views

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
votes

0
answers

117
views

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
votes

0
answers

82
views

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

0
answers

69
views

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
votes

0
answers

67
views

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
votes

1
answer

262
views

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 $n$ ...

0
votes

0
answers

42
views

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
votes

0
answers

108
views

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

1
answer

227
views

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
votes

0
answers

26
views

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

1
answer

154
views

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

1
answer

372
views

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
votes

0
answers

123
views

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

1
answer

365
views

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
votes

0
answers

188
views

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

1
answer

111
views

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

0
answers

78
views

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

0
answers

117
views

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

1
answer

242
views

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(...