Banach Lattices Research Articles

Positively limited sets in Banach lattices

We introduce and study the class of positively limited sets in Banach lattices, that is, sets on which every weak⁎ null sequence of positive functionals converges uniformly to zero. Moreover the relationships between the class of positively limited sets and other known classes of sets such as almost limited sets and (weakly) compact sets are discussed. Some properties of Banach lattices can be characterized in terms of positively limited sets.

Nuclear operators and applications to kernel operators

AbstractLet Σ be a σ‐algebra of subsets of a set Ω and be the Banach lattice of bounded Σ‐measurable real functions on Ω. For a Banach space E, we establish the relationship between a countably additive measure of finite variation with a ‐Bochner integrable derivative and nuclearity of the corresponding integration operator . As an application, we derive that if Ω is a topological Hausdorff space and Y is a compact Hausdorff space and , then the corresponding kernel operator is nuclear.

Free complex Banach lattices

The construction of the free Banach lattice generated by a real Banach space is extended to the complex setting. It is shown that for every complex Banach space E there is a complex Banach lattice FBLC[E] containing a linear isometric copy of E and satisfying the following universal property: for every complex Banach lattice XC, every operator T:E→XC admits a unique lattice homomorphic extension Tˆ:FBLC[E]→XC with ‖Tˆ‖=‖T‖. The free complex Banach lattice FBLC[E] is shown to have analogous properties to those of its real counterpart. However, examples of non-isomorphic complex Banach spaces E and F can be given so that FBLC[E] and FBLC[F] are lattice isometric. The spectral theory of induced lattice homomorphisms on FBLC[E] is also explored.

Order intervals in $C(K)$. Compactness, coincidence of topologies, metrizability

Let $K$ be a compact space and let $C(K)$ be the Banach lattice of real-valued continuous functions on $K$. We establish eleven conditions equivalent to the strong compactness of the order interval $[0,x]$ in $C(K)$, including the following ones: (i) $\{x>0\}$ consists of isolated points of $K$; (ii) $[0,x]$ is pointwise compact; (iii) $[0,x]$ is weakly compact; (iv) the strong topology and that of pointwise convergence coincide on $[0,x]$; (v) the strong and weak topologies coincide on $[0,x]$. \noindent Moreover, the weak topology and that of pointwise convergence coincide on $[0,x]$ if and only if $\{x>0\}$ is scattered. Finally, the weak topology on $[0,x]$ is metrizable if and only if the topology of pointwise convergence on $[0,x]$ is such if and only if $\{x>0\}$ is countable.

A Nakano carrier theorem for polynomials

We use a localisation technique to study orthogonally additive polynomials on Banach lattices. We derive alternative characterisations for orthogonal additivity of polynomials and orthosymmetry of m m -linear mappings. We prove that an orthogonally additive polynomial which is order continuous at one point is order continuous at every point and we give an example to show that this result does not extend to regular polynomials in general. Finally, we prove a Nakano Carrier theorem for orthogonally additive polynomials, generalising a result of Kusraev [Orthosymmetric bilinear operators, Vladikavkaz, preprint, 2007].

The Property (D) and the Almost Limited Completely Continuous Operators

In this paper, we study some geometric properties in Banach lattices and the class of almost limited completely continuous operators. For example, we study Banach lattices with property (d) and we give a new characterization of this property in terms of the solid hull of almost limited sets.

Automatic continuity of separating and biseparating isomorphisms

Let p p be a fixed number with 1 ≤ p > ∞ 1 \leq p > \infty . It is shown that every surjective and biseparating linear map between L p L^p -spaces is continuous when the underlying measure space is non-atomic. We also prove that a separating isomorphism on l p l^p is both continuous and biseparating. Furthermore, these (bi-)separating maps take the form of a weighted composition operator. Our proofs are direct, elementary and do not invoke deep results about Riesz spaces or Banach lattices.

Korovkin-Type Theorems for Weakly Nonlinear and Monotone Operators

In this paper we prove analogs of Korovkin’s theorem in the context of weakly nonlinear and monotone operators acting on Banach lattices of functions of several variables. Our results concern the convergence almost everywhere, the convergence in measure and the convergence in \(L^{p}\)-norm. Several results illustrating the theory are also included.

Weak fixed point property for a Banach lattice of some compact operator spaces

Weak fixed point property for a Banach lattice of some compact operator spaces

The double transpose of the Ruelle operator

In this paper we study the double transpose of the $$L^1(X,\mathscr {B}(X),\nu )$$ -extensions of the Ruelle transfer operator $$\mathscr {L}_{f}$$ associated to a general real continuous potential $$f\in C(X)$$ , where $$X=E^{\mathbb {N}}$$ , the alphabet E is any compact metric space and $$\nu $$ is a maximal eigenmeasure. For this operator, denoted by $$\mathbb {L}^{**}_{f}$$ , we prove the existence of some non-negative eigenfunction, in the Banach lattice sense, associated to $$\rho (\mathscr {L}_{f})$$ , the spectral radius of the Ruelle operator acting on C(X). As an application, we obtain a sufficient condition ensuring that the extension of the Ruelle operator to $$L^1(X,\mathscr {B}(X),\nu )$$ has an eigenfunction associated to $$\rho (\mathscr {L}_{f})$$ . These eigenfunctions agree with the usual maximal eigenfunctions, when the potential f belongs to the Hölder, Walters or Bowen class. We also construct solutions to the classical and generalized variational problem, using the eigenvector constructed here.

The strong BD property in Banach lattices

We introduce the concept of strong BD property in Banach lattices and we characterize Banach lattices with this property. Also, by introducing the class of almost limited (resp. Dunford-Pettis) weakly completely continuous operators from an arbitrary Banach lattice E to another F, we give some properties of them related to some well known classes of operators and related to the strong BD (resp. RDP?) property of the Banach lattice E.

On regular operators on Banach lattices

Let $E$ and $F$ be Banach lattices and $X$ and $Y$ be Banach spaces. A linear operator $T: E \rightarrow F$ is called regular if it is the difference of two positive operators. $L_{r}(E,F)$ denotes the vector space of all regular operators from $E$ into $F$. A continuous linear operator $T: E \rightarrow X$ is called $M$-weakly compact operator if for every disjoint bounded sequence $(x_{n})$ in $E$, we have $lim_{n \rightarrow\infty} \| Tx_{n} \| =0$. $W^{r}_{M}(E,F)$ denotes the regular $M$-weakly compact operators from $E$ into $F$. This paper is devoted to the study of regular operators and $M$-weakly compact operators on Banach lattices. We show that $F$ has a b-property if and only if $L_{r}(E,F)$ has b-property. Also, $W^{r}_{M}(E,F)$ is a $KB$-space if and only if $F$ is a $KB$-space.

The strong Dunford-Pettis relatively compact property of order p

We introduce and study Banach lattices with the strong Dunford-Pettis relatively compact property of order p (1 ? p < ?); that is, spaces in which every weakly p-compact and almost Dunford-Pettis set is relatively compact. We also introduce the notion of the weak Dunford-Pettis property of order p and then characterize this property in terms of sequences. In particular, in terms of disjoint weakly compact operators into c0, an operator characterization of those Banach lattices with the weak Dunford-Pettis property of order p is given. Moreover, some results about Banach lattices with the positive Dunford-Pettis relatively compact property of order p are presented

The class of demi KB-operators on Banach lattices

In this paper, we introduce and study the new concept of demi KB-operators. Let $E$ be a Banach lattice. An operator $T: E\longrightarrow E$ is said to be a demi KB-operator if, for every positive increasing sequence $\{x_{n}\}$ in the closed unit ball $\mathcal{B}_{E}$ of $E$ such that $\{x_{n}-Tx_{n}\}$ is norm convergent to $x\in E$, there is a norm convergent subsequence of $\{x_{n}\}$. If the latter sequence has a weakly convergent subsequence then $T$ is called a weak demi KB-operator. We also investigate the relationship of these classes of operators with classical notions of operators, such as b-weakly demicompact operators and demi Dunford-Pettis operators.

On unbounded norm demi KB-operators on Banach lattices

On unbounded norm demi KB-operators on Banach lattices

Nuclear operators and Baire vector measures

Let $C_b(X)$ be the Banach lattice of all bounded continuous real functions on a completely regular Hausdorff space $X$ and let $(E,\|\cdot\|_E)$ be a real Banach space. A linear operator $T:C_b(X)\to E$ is said to be $\sigma$-additive if $\|T(u_n)\|_E\to 0$ whenever $(u_n)$ is a sequence in $C_b(X)$ such that $u_n(x)\downarrow 0$ for each $x\in X$. We characterize $\sigma$-additive nuclear operators $T:C_b(X)\to E$ in terms of their representing Baire vector measures. The formula for the trace of $\sigma$-additive nuclear operators $T:C_b(X)\to C_b(X)$ is derived.

Growth rate of eventually positive Kreiss bounded $$C_0$$-semigroups on $$L^p$$ and $$\mathcal {C}(K)$$

In this paper, we compare several Cesàro- and Kreiss-type boundedness conditions for a \(C_0\)-semigroup on a Banach space and we show that those conditions are all equivalent for a positive semigroup on a Banach lattice. Furthermore, we give an estimate of the growth rate of a Kreiss bounded and eventually positive \(C_0\)-semigroup \((T_t)_{t\ge 0}\) on certain Banach lattices X. We prove that if X is an \(L^p\)-space, \(1<p<+\infty \), then \(\Vert T_t\Vert = \mathcal {O}\left( t/\log (t)^{\max (1/p,1/p')}\right) \), and if X is an \((\text {AL})\) or \((\text {AM})\)-space, then \(\Vert T_t\Vert =\mathcal {O}(t^{1-\epsilon })\) for some \(\epsilon \in (0,1)\), improving previous estimates.

Quantitative positive Schur property in Banach lattices

We introduce and investigate quantitative versions of the positive Schur property. We show, in particular, that each L 1 ( μ ) L_{1}(\mu ) -space( μ \mu σ \sigma -finite) enjoys the 1 1 -positive Schur property in the strongest possible way. By introducing a quantity measuring L L -weak non-compactness of sets in Banach lattices, we prove that the positive Schur property is always quantitative. We also show that this quantity is equal to several natural measures of weak non-compactness in abstract L L -spaces.

Semilinear elliptic equations on rough domains

The paper makes use of recent results in the theory of Banach lattices and positive operators to deal with abstract semilinear equations. The aim is to work with minimal or no regularity conditions on the boundary of the domains, where the usual arguments based on maximum principles do not apply. A key result is an application of Kato's inequality to prove a comparison theorem for eigenfunctions that only requires interior regularity and avoids the use of the Hopf boundary maximum principle. We demonstrate the theory on an abstract degenerate logistic equation by proving the existence, uniqueness and stability of non-trivial positive solutions. Examples of operators include the Dirichlet Laplacian on arbitrary bounded domains, a simplified construction of the Robin Laplacian on arbitrary domains with boundary of finite measure and general elliptic operators in divergence form.

Complete latticeability in vector‐valued sequence spaces

AbstractUsing a technique due to Jiménez–Rodríguez, first we prove the complete latticeability of the set of disjoint non‐norm null weakly null sequences and of the set of disjoint non‐norm null regular‐polynomially null sequences in Banach lattices. Then we apply the mother vector technique to prove the complete latticeability of , which implies the complete latticeability of , where E is a Banach lattice and .