Fatou Property Research Articles

BALAYAGE MONOTONOUS RISK MEASURES

We consider coherent risk measures satisfying the Fatou property which are monotonous with respect to balayage or dilatation. An equivalent condition ensuring balayage-monotonicity is given and a representation result is derived.

On Interpolation in LpSpaces

then X is an interpolation space between the spaces Lp and Lq . This means that any bounded linear operator T in Lp and Lq is also bounded in the space X . This theorem was proved by Boyd [1] in 1967 under the additional assumption that X possesses the Fatou property (see also [2] and [3, Theorem 5.16]). For the case in which the space X is separable or possesses the Fatou property, see [4, Theorem 2.b.11]. Combining estimates for the Calderon maximum operator and Semenov’s Lemma [5, Chap. 2, Lemma 4.7], we can prove Boyd’s theorem for arbitrary symmetric spaces. This result is also contained implicitly in Theorem 6.12 from [5, Chap. 2]. In the case q = ∞ , the assumption in Boyd’s theorem can be weakened. Namely, the fact that X is an interpolation space between Lp and L∞ is a consequence solely of the one-sided estimate

Positivity Preserving Forms have the Fatou Property

We prove that if (e,De) is a positivity preserving form on L 2 (E;m), and if (u n)n∈ℕ is a sequence in D(e) converging m-almost everywhere to u ∈ L 2 (E;m), then e (u,u) ⩽ lim infn e(u n ,u n ).

The dual space of L∞ is L1

The provocative title may be supposed to hold if the axiom of choice is weakened. Even more: The dual space of any Köthe space (real-valued and over a σ-finite measure space) coincides with its associate space in the canonical way; in particular, any Köthe space with Fatou's property is re-flexive. We will show that all this is true in ZF + DC + PM ω, where PM ω is introduced in the text and holds e.g. in Solovay's model.

Lipschitz Continuity of the Absolute Value and Riesz Projections in Symmetric Operator Spaces

A principal result of the paper is that ifEis a symmetric Banach function space on the positive half-line with the Fatou property then, for all semifinite von Neumann algebras (M, τ), the absolute value mapping is Lipschitz continuous on the associated symmetric operator spaceE(M, τ) with Lipschitz constant depending only onEif and only ifEhas non-trivial Boyd indices. It follows that if M is any von Neumann algebra, then the absolute value map is Lipschitz continuous on the corresponding HaagerupLp-space, provided 1<p<∞.

PARTIALLY ORDERED NORMED lINEAR SPACES WITH WEAK FATOU PROPERTY

Let $E$ be a Riesz space with lattice ordered norm $\|\cdot\|$. Amemiya proved that $E$ is complete under this norm if $E$ has weak Fatou property for monotone sequence (: $E$ is monotone complete) with respect to the norm $\|\cdot\|$. This is a generalization of the Riesz-Fisher's and the Nakano's theorem. In the cases of non normed Riesz space or non lattice ordered norm, this theorem is not true in general. We shall investigate in this paper a necessary and sufficient condition for Amemiya's theorem to be valid in a partially ordered normed linear space.

Fatou property of harmonic maps from complete manifolds with simple type ends

Fatou property of harmonic maps from complete manifolds with simple type ends

PARTIALLY ORDERED NORMED lINEAR SPACES WITH WEAK FATOU PROPERTY

Let $E$ be a Riesz space with lattice ordered norm $\|\cdot\|$. Amemiya proved that $E$ is complete under this norm if $E$ has weak Fatou property for monotone sequence (: $E$ is monotone complete) with respect to the norm $\|\cdot\|$. This is a generalization of the Riesz-Fisher's and the Nakano's theorem. In the cases of non normed Riesz space or non lattice ordered norm, this theorem is not true in general. We shall investigate in this paper a necessary and sufficient condition for Amemiya's theorem to be valid in a partially ordered normed linear space.

Some consequences of a Fatou property of the tropical semiring

We show that the equatorial semiring ▪ is a Fatou extension of the tropical semiring ▪. This property allows us to give partial decidability results for the equality problem for rational series with multiplicities in the tropical semiring. We also deduce from it the decidability of the limitedness problem for the equatorial semiring, solving therefore a question of I. Simon.

On a kind of Fatou property of context-free groups

We show that an algebraic subset of a context-free group G which is included in a subgroup U of G is algebraic in U.

Quojections and finite-dimensional decompositions

K. Ftoret and V. B. Moscatelli proved in [5] that a Fr6chet space with an unconditional basis and without a continuous norm decomposes as a product of Fr6chet spaces with unconditional bases and continuous norms. This implies that a quojection with an unconditional basis is isomorphic to a product of Banach spaces with unconditional bases. In the same direction J. C. Diaz showed that no twisted quojection can have a complete lattice structure with the (weak)-Fatou property (cf. [2]). On the other hand, there are twisted quojections with a (conditional) basis, as shown in [8]. We present here an example of a twisted quojection with an unconditional finite-dimensional decomposition (FDD) in subspaces of dimension 2 (2-FDD in short) and with a basis. The method also applies to nuclear spaces and allows us to produce a nuclear Fr~chet space with a 2 -FDD which is not the product of spaces with continuous norms. This is done in Section 2. In Section 3 we show that a quojection with a strong F D D has a basis. For further information about quojections see the survey [9]. The notation here used is standard (cf. [6]) and we recall only some facts about finite-dimensional decompositions. Let F be a Fr6chet space and let (F,) be a sequence of closed subspaees of F. (F,) is a Schauder decomposition o f F if for every x in F there is a unique expansion x = 52 x,, with n x, e F,. If F, is finite-dimensional for any n, we say that (F,) is a finite-dimensional decomposition of F (FDD). We say that (F,) is a strong finite-dimensional decomposition of F (SFDD) if r = supdim(F,) < oe and, in this case, we say that (F.) is a r -FDD of F. If in the above definitions we require the unconditional convergence of the series x = Z x., we speak of unconditional Schauder decompositions, unconditional F D D and n unconditional SFDD.

On the modulus of cone absolutely summing operators and vector measures of bounded variation

Let E E and F F be Banach lattices. It is shown that if F F has the Levi and the Fatou property, then the ordered Banach space L l ( E , F ) {\mathcal {L}^l}\left ( {E,F} \right ) of cone absolutely summing operators is a Banach lattice and an order ideal of the Riesz space L r ( E , F ) {\mathcal {L}^r}\left ( {E,F} \right ) of regular operators. The same argument yields a Jordan decomposition of F F -valued vector measures of bounded variation.

A characterization of dual Banach lattices

In this paper we give a characterization of dual Banach lattices. In fact, we prove that a Banach function space E on a separable measure space which has the Fatou property is a dual Banach lattice if and only if all positive operators from L 1(0,1) into E are abstract kernel operators, hence extending the fact, proved by M. Talagrand, that separable Banach lattices with the Radon-Nikodym property are dual Banach lattices.

On Levi-Like Properties and some of Their Applications in Riesz Space Theory

AbstractLet (L, λ) be a locally solid Riesz space. (L, λ) is said to have the Levi property if for every increasing λ-bounded net (xα) ⊂ L+, sup xα exists. The Levi property, appearing in literature also as weak Fatou property (Luxemburg and Zaanen), condition (B) or monotone completeness (Russian terminology), is a classical object of investigation. In this paper we are interested in some variations of the property, their mutual relationships and applications in the theory of topological Riesz spaces. In the first part of the paper we clarify the status of two problems of Aliprantis and Burkinshaw. In the second part we study ideal-injective Riesz spaces.

Factorization in Banach function spaces

Let ϱ be a function norm based on a σ-finite measure space (Ω,Σ,μ). The Hölder inequality implies that, given f in L ϱ and g in L ϱ, the product fg belongs to L 1( μ) and satisfies | fg|1≤ ϱ( f) ϱ′( g). It is proved that, if ϱ is saturated and has the Fatou property, then every function ϕ in L 1( μ) can be factorized as ϕ = fg, with f in L ϱ, g in L ϱ′ and | ϕ|1= ϱ( f) ϱ′( g). If ϱ is saturated and has the Riesz-Fischer property, then every L 1-function can still be factorized in this way, but the condition on the norms of the factors must be weakened. A further weakening of the conclusion is necessary for the case when ϱ is assumed merely to be saturated. In that case, such a factorization of an arbitrary L 1-function may not be possible, although it is almost possible in a certain measure-theoretic sense.

Cyclic Banach spaces and reflexive operator algebras

SynopsisLet X be a Banach space and let ℬ be a σ-complete Boolean algebra of projections on X with a cyclic vector. It is shown that there exists a normed Köthe space Lρ, the norm of which has the Fatou property, such that X is linearly homeomorphic to the subspace of Lρ consisting of those functions of absolutely continuous norm and such that, under this homeomorphism, the projections ℬ correspond to operators consisting of multiplication by characteristic functions. This representation theorem for X is used to show that certain operator algebras associated with ℬ are reflexive. As an immediate corollary of the reflexivity result, it is shown that, if T is a scalar type spectral operator whose resolution of the identity has a cyclic vector, then T is reflexive.

Riesz Seminorms with Fatou Properties

A seminormed Riesz space ${L_\rho }$ satisfies the $\sigma$-Fatou property (resp. the Fatou property) if $\theta \leq {u_n} \uparrow u$ in $L$ (resp. $\theta \leq {u_\alpha } \uparrow u\;{\text {in}}\;L$) implies $\rho ({u_n}) \uparrow \rho (u)$ (resp. $\rho ({u_\alpha }) \uparrow \rho (u)$). The following results are proved: (i) A normed Riesz space ${L_\rho }$ satisfies the $\sigma$-Fatou property if, and only if, its norm completion does and ${L_\rho }$ has $({\mathbf {A}},0)$. (ii) The quotient space ${L_\rho }/{I_\rho }$ has the Fatou property if ${L_\rho }$ is Archimedean with the Fatou property. $({I_\rho } = \{ u\varepsilon L:\rho (u) = 0\} .)$ (iii) If ${L_\rho }$ is almost $\sigma$-Dedekind complete with the $\sigma$-Fatou property, then ${L_\rho }/{I_\rho }$ has the $\sigma$-Fatou property. A counterexample shows that (iii) may be false for Archimedean Riesz spaces.

Riesz seminorms with Fatou properties

A seminormed Riesz space L ρ {L_\rho } satisfies the σ \sigma -Fatou property (resp. the Fatou property) if θ ≤ u n ↑ u \theta \leq {u_n} \uparrow u in L L (resp. θ ≤ u α ↑ u in L \theta \leq {u_\alpha } \uparrow u\;{\text {in}}\;L ) implies ρ ( u n ) ↑ ρ ( u ) \rho ({u_n}) \uparrow \rho (u) (resp. ρ ( u α ) ↑ ρ ( u ) \rho ({u_\alpha }) \uparrow \rho (u) ). The following results are proved: (i) A normed Riesz space L ρ {L_\rho } satisfies the σ \sigma -Fatou property if, and only if, its norm completion does and L ρ {L_\rho } has ( A , 0 ) ({\mathbf {A}},0) . (ii) The quotient space L ρ / I ρ {L_\rho }/{I_\rho } has the Fatou property if L ρ {L_\rho } is Archimedean with the Fatou property. ( I ρ = { u ε L : ρ ( u ) = 0 } . ) ({I_\rho } = \{ u\varepsilon L:\rho (u) = 0\} .) (iii) If L ρ {L_\rho } is almost σ \sigma -Dedekind complete with the σ \sigma -Fatou property, then L ρ / I ρ {L_\rho }/{I_\rho } has the σ \sigma -Fatou property. A counterexample shows that (iii) may be false for Archimedean Riesz spaces.