# Questions tagged [banach-lattices]

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

25 questions
Filter by
Sorted by
Tagged with
84 views

### 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:...
177 views

### 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 $X$ ...
280 views

### 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 ...
198 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 ...
148 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 ...
145 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 ...
193 views

73 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 ...
62 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 ...
67 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 $L_1(\mu)$...
256 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 $n$ ...
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 ...
94 views

154 views

308 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 $f,h \in L^1$ ...
178 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 ...
104 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) ...
75 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 ...
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 ...
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(...