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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Does integration by parts formula hold in $H^1(0,T,L^2(\Omega))$?

Let $\Omega$ be an open set from $\mathbb{R}^N$. How can we prove that if $u,v\in H^1(0,T,L^2(\Omega))$ (in Bochner sense) then $(u\cdot v)'\in L^2(0,T,L^1(\Omega))$ with $(uv)'=u'v+v'u$ and the ...
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Weak$^\ast$ closure of a countably complete sublattice of the unit ball of $L^\infty(\Omega, \mu)$

This is a reframing of my previous question from a Banach lattice perspective: Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity? The previous question ...
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Approximating continuous functions from $K\times L$ into $[0,1]$

Let $K$ and $L$ be compact Hausdorff spaces, let $f:K\times L\to [0,1]$ be continuous and let $\varepsilon>0$. Can we find continuous $g_{1},...,g_{n}:K\to[0,+\infty)$ and $h_{1},...,h_{n}:L\to[0,+\...
erz's user avatar
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interiors of positive cones in ordered Banach spaces

I have a couple of questions about ordered Banach spaces and interiors of their positive cones. I would appreciate your insights and any recommended references. I want to know several examples of ...
Saito's user avatar
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Pointwise convergence and disjoint sequences in $C(K)$

Let $K$ be a Hausdorff compact space and let $C(K)$ be the space of continuous real-valued functions on $K$. A sequence $(h_n)$ in $C(K)$ is called almost disjoint if there is a sequence $(g_n)$ with ...
erz's user avatar
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Banach tori: classification up to Fréchet homeomorphisms

Consider the set $T$ in $l_p$ defined as closure of \begin{equation} T = \{ (x_1,\dotsc,x_n,\dotsc): x_j = \frac{1}{2^{(j/p)}} e^{it_j}, t_j \in \mathbb{R}/\mathbb{Z} \}. \end{equation} This seems to ...
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Decomposition of weak* convergent nets into positive weak* convergent nets

Let $F$ be an order unit Banach space with order unit $e$ and topological dual space $F^*$ ordered by the dual cone. Let $E\subset F^*$ be a closed subspace that separates points of $F$ and such that ...
Nick's user avatar
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Convex combination of positive mean-ergodic operators

Let $T_1,T_2:L^1([0,1],\mathrm{d}x)\to L^1([0,1],\mathrm{d}x)$ be positive mean-ergodic operators such that: For every $h:[0,1]\to \mathbb{R}_+$ we have that $$\int_0^1 T_1 h(x)\mathrm{d}x = \int_0^1 ...
Matheus Manzatto's user avatar
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Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace

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 $\...
Jochen Glueck's user avatar
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The initial sigma-algebra on the dual of a Banach lattice

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$^*$-...
HardyHulley's user avatar
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Is the Lorentz space $L_{W,1}(0,1)$ isomorphic to $L_1(0,1)$?

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 ...
Antonio Martinon 's user avatar
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The embedding of a Banach lattice in an ultrapower

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}(...
M.González's user avatar
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The complement of $L_1(0,1)$ in $L_1(0,1)^{**}$

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(\...
M.González's user avatar
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Can we decompose an increasing net of functions into two increasing nets with prescribed supports?

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 $f_{i}$ ...
erz's user avatar
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quasi-Banach function spaces are subspace of $L_p$

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 ...
user92646's user avatar
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Eigenvectors of the dual of positive irreducible operators

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 ...
Matheus Manzatto's user avatar
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Is a certain property of a continuous map preserved under "surjectification"?

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 $...
erz's user avatar
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Reference on inductive (direct) limit of ordered vector spaces and vector lattices

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 ...
erz's user avatar
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Looking for a paper on axiomatic orthogonality in a vector space

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 ...
erz's user avatar
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How to characterize the order convergence in Bochner-integrable functions space?

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:...
grutzchell's user avatar
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Is the union of good equivalence relations on a compact space good?

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 $X$ ...
erz's user avatar
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4 votes
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When do positive operators have eigenvalues?

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 ...
Constantin K's user avatar
9 votes
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260 views

Is a Banach lattice isomorphic to a Hilbert space in fact a Hilbert lattice?

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 ...
erz's user avatar
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6 votes
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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 ...
Yury Korolev's user avatar
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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 ...
Mark Roelands's user avatar
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2 answers
399 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}:...
Giuliosky's user avatar
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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 ...
Pietro Majer's user avatar
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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 ...
Mark's user avatar
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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 ...
Jochen Glueck's user avatar
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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 $L_1(\mu)$...
M.González's user avatar
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4 votes
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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 $n$ ...
M.González's user avatar
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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 ...
0xbadf00d's user avatar
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$\ell^\infty / ces_0$ 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^\...
Miek Messerschmidt's user avatar
5 votes
1 answer
249 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 ...
Ben W's user avatar
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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 \...
James Hanson's user avatar
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3 votes
1 answer
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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) \...
Mark's user avatar
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3 votes
1 answer
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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 ...
Mark's user avatar
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3 votes
0 answers
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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 $...
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5 votes
1 answer
452 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 $f,h \in L^1$ ...
Jochen Glueck's user avatar
5 votes
0 answers
233 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 ...
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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) ...
encore's user avatar
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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 ...
Fred Dashiell's user avatar
2 votes
0 answers
119 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 ...
user786's user avatar
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2 votes
1 answer
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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(...
Shinning Star's user avatar