We introduce UDSp-property (resp. UDTq-property) in Banach lattices as the property that every normalized disjoint sequence has a subsequence with an upper p-estimate (resp. lower q-estimate). In the case of rearrangement invariant spaces, the relationships with Boyd indices of the space are studied. Some applications of these properties are given to the high order smoothness of Banach lattices, in the sense of the existence of differentiable bump functions.
We introduce and study a new class of operators that we call disjoint weak Banach-Saks operators. We establish some characterizations of this class of operators by different types of convergence (norm convergence, unbounded order convergence, unbounded norm convergence and unbounded absolute weak convergence) as well as by the positive weakly null sequences. Consequently, we give a new characterization of the disjoint weak Banach-Saks property by the positive disjoint weakly null sequences. Furthermore, we study the relationship between this class and other classes of operators.
This is a survey on disjointly homogeneous Banach lattices and their applicactions. Several structural properties of this class are analyzed. In addition we show how these spaces provide a natural framework for studying the compactness of powers of operators allowing for a unified treatment of well-known results.
An operator $T$ from a Banach lattice $E$ into a Banach space is disjointly non-singular ($DN$-$S$, for short) if no restriction of $T$ to a subspace generated by a disjoint sequence is strictly singular. We obtain several results for $DN$-$S$ operators, including a perturbative characterization. For $E=L_p$ ($1< p<\infty$) we improve the results, and we show that the $DN$-$S$ operators have a different behavior in the cases $p=2$ and $p\neq 2$. As an application we prove that the strongly embedded subspaces of $L_p$ form an open subset in the set of all closed subspaces.
We generalize results concerning $\mathrm{C}_0$-semigroups on Banach lattices to a setting of ordered Banach spaces. We prove that the generator of a disjointness preserving $\mathrm{C}_0$-semigroup is local. Some basic properties of local operators are also given. We investigate cases where local operators generate local $\mathrm{C}_0$-semigroups, by using Taylor series or Yosida approximations. As norms we consider regular norms and show that bands are closed with respect to such norms. Our proofs rely on the theory of embedding pre-Riesz spaces in vector lattices and on corresponding extens…
Let C be an axiomatizable class of order continuous real or complex Banach lattices, that is, this class is closed under isometric vector lattice isomorphisms and ultraproducts, and the complementary class is closed under ultrapowers. We show that if linear isometric embeddings of members of C in their ultrapowers preserve disjointness, the class C^B of Banach spaces obtained by forgetting the Banach lattice structure is still axiomatizable. Moreover if C coincides with its "script class" SC, so does C^B with SC^B. This allows us to give new examples of axiomatizable classes of Banach spaces, …
In this article we investigate the disjointly non-singular (DNS) operators. Following [8] we say that an operator $T$ from a Banach lattice $F$ into a Banach space $E$ is DNS, if no restriction of $T$ to a subspace generated by a disjoint sequence is strictly singular. We partially answer a question from [8] by showing that this class of operators forms an open subset of $L\left(F,E\right)$ as soon as $F$ is order continuous. Moreover, we show that in this case $T$ is DNS if and only if the norm topology is the minimal topology which is simultaneously stronger than the unbounded norm topology …
We describe essential (in particular Fredholm and semi-Fredholm) spectra of operators on Banach lattices of the form $T=wU$, where $w$ is a central operator and $U$ is a disjointness preserving operator such that its spectrum $σ(U)$ is a subset of the unit circle.
We explore the relation between lattice versions of strict singularity for operators from a Banach lattice to a Banach space. In particular, we study when the class of disjointly strictly singular operators, those not invertible on the span of any disjoint sequence, coincides with that of lattice strictly singular operators, i.e. those not invertible on any (infinite dimensional) sublattice. New results are given which help to clarify the existing relation between these two classes.
Several recent papers were devoted to various modifications of limited, Grothendieck, and Dunford--Pettis operators, etc., through involving the Banach lattice structure. In the present paper, it is shown that many of these operators appear as operators affiliated to well known properties of Banach lattices, like the disjoint (dual) Schur property, the disjoint Grothendieck property, the property (d), and the sequential w*-continuity of the lattice operations. It is proved that the spaces consisting of regularly versions of the above operators are all Banach spaces. The domination problem for …
We give a characterization of conditional expectation operators through a disjointness type property similar to band preserving operators. We say that the operator $T:X\to X$ on a Banach lattice $X$ is semi band preserving if and only if for all $f, g \in X$, $f \perp Tg$ implies that $Tf \perp Tg$. We prove that when $X$ is a purely atomic Banach lattice, then an operator $T$ on $X$ is a weighted conditional expectation operator if and only if $T$ is semi band preserving.
We consider uncountable almost disjoint families of subsets of $\mathbb N$, the Johnson-Lindenstrauss Banach spaces $(\mathcal X_{\mathcal A}, \|\ \|_\infty)$ induced by them, and their natural equivalent renormings $(\mathcal X_{\mathcal A}, \|\ \|_{\infty, 2})$. We introduce a partial order $\mathbb P_{\mathcal A}$ and characterize some geometric properties of the spheres of $(\mathcal X_{\mathcal A}, \|\ \|_{\infty})$ and of $(\mathcal X_{\mathcal A}, \|\ \|_{\infty, 2})$ in terms of combinatorial properties of $\mathbb P_{\mathcal A}$. Exploiting the extreme behavior of some known and some…
A Banach lattice E is called p-disjointly homogeneous, 1< p< infty, when every sequence of pairwise disjoint normalized elements in E has a subsequence equivalent to the unit vector basis of l_p. Employing methods from interpolation theory, we clarify which rearrangement invariant (r.i.) spaces on [0,1] are p-disjointly homogeneous. In particular, for every 1<p< infty and any increasing concave function f on [0,1], which is not equivalent neither 1 nor t, there exists a p-disjointly homogeneous r.i. space with the fundamental function f. Moreover, in the class of all interpolation r.i. spaces …
We study the stability of disjointness preservers on Banach lattices. In many cases, we prove that an "almost disjointness preserving" operator is well approximable by a disjointess preserving one. However, this approximation is not always possible, as our examples show.
Generalized almost disjoint families and injective Banach spaces
Institute of Mathematics, Czech Academy of Sciences（捷克科学院数学研究所）；Institute for Advanced Study in Mathematics, Harbin Institute of Technology（哈尔滨工业大学高等研究院）
A pair of functions defined on a set X with values in a vector space E is said to be disjoint if at least one of the functions takes the value 0 at every point in X. An operator acting between vector-valued function spaces is disjointness preserving if it maps disjoint functions to disjoint functions. We characterize compact and weakly compact disjointness preserving operators between spaces of Banach space-valued differentiable functions.
We introduce several different notions of disjoint distributional chaos for sequences of multivalued linear operators in Fréchet spaces. Any of these notions seems to be new and not considered elsewhere even for linear continuous operators in Banach spaces. We focus special attention to the analysis of some specific classes of linear continuous operators having a certain disjoint distributionally chaotic behaviour, providing also a great number of illustrative examples and applications of our abstract theoretical results.
Let $1\le p<\infty$. A symmetric space $X$ on $[0,1]$ is said to be $p$-disjointly homogeneous (resp. restricted $p$-disjointly homogeneous) if every sequence of normalized pairwise disjoint functions from $X$ (resp. characteristic functions) contains a subsequence equivalent in $X$ to the unit vector basis of $l_p$. Answering a question posed recently, we construct, for each $1\le p<\infty$, a restricted $p$-disjointly homogeneous symmetric space, which is not $p$-disjointly homogeneous. Moreover, we prove that the property of $p$-disjoint homogeneity is preserved under Banach isomorphisms.
A Banach space $E$ is said to be injective if for every Banach space $X$ and every subspace $Y$ of $X$ every operator $t:Y\to E$ has an extension $T:X\to E$. We say that $E$ is $\aleph$-injective (respectively, universally $\aleph$-injective) if the preceding condition holds for Banach spaces $X$ (respectively $Y$) with density less than a given uncountable cardinal $\aleph$. We perform a study of $\aleph$-injective and universally $\aleph$-injective Banach spaces which extends the basic case where $\aleph=\aleph_1$ is the first uncountable cardinal. When dealing with the corresponding "isomet…
The disjointly non-singular ($DNS$) operators $T\in L(E,Y)$ from a Banach lattice $E$ to a Banach space $Y$ are those operators which are strictly singular in no closed subspace generated by a disjoint sequence of non-zero vectors. When $E$ is order continuous with a weak unit, $E$ can be represented as a dense ideal in some $L_1(μ)$ space, and we show that each of $T\in DNS(E,Y)$ admits an extension $\overline{T}\in DNS(L_1(μ),PO)$ from which we derive that both $T$ and $T^{**}$ are tauberian operators and that the operator $T^{co}: E^{**}/E\to Y^{**}/Y$ induced by $T^{**}$ is an (into) isomo…
We prove that the class of Banach function lattices in which all relatively weakly compact sets are equi-integrable sets (i.e. spaces satisfying the Dunford-Pettis criterion) coincides with the class of 1-disjointly homogeneous Banach lattices. A new examples of such spaces are provided. Furthermore, it is shown that Dunford-Pettis criterion is equivalent to de la Vallee Poussin criterion in all rearrangement invariant spaces on the interval. Finally, the results are applied to characterize weakly compact pointwise multipliers between Banach function lattices.
Let $I$ be a countably infinite index set, and let $X$ be a Banach sequence space over $I.$ In this article, we characterize disjoint hypercyclic and supercyclic weighted pseudo-shift operators on $X$ in terms of the weights, the OP-basis, and the shift mappings on $I.$ Also, the shifts on weighted $L^p$ spaces of a directed tree and the operator weighted shifts on $\ell^2(\mathbb{Z,\mathcal{K}})$ are investigated as special cases.
First we adjust a technique due to Jiménez-Rodríguez to 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 $λ_π(E) \setminus λ_s(E)$, which implies the complete latticeability of $(\ell_p\widehat{\otimes}_{|π|}E)\setminus(\ell_p\widehat{\otimes}_πE)$, where $E$ is a Banach lattice and $1 < p < \infty$.
We study Hahn-Banach extensions of multilinear forms defined on Banach sequence spaces. We characterize $c_0$ in terms of extension of bilinear forms, and describe the Banach sequence spaces in which every bilinear form admits extensions to any superspace.
Let $1\le p<\infty$. A Banach lattice $E$ is said to be disjointly homogeneous (resp. $p$-disjointly homogeneous) if two arbitrary normalized disjoint sequences from $E$ contain equivalent in $E$ subsequences (resp. every normalized disjoint sequence contains a subsequence equivalent in $E$ to the unit vector basis of $l_p$). Answering a question raised in 2014 by Flores, Hernandez, Spinu, Tradacete, and Troitsky, for each $1<p<\infty$, we construct a reflexive $p$-disjointly homogeneous rearrangement invariant space on $[0,1]$ whose dual is not disjointly homogeneous. Employing methods from i…
The new concepts are introduced of almost overcomplete sequence in a Banach space and almost overtotal sequence in a dual space. We prove that any of such sequences is relatively norm-compact and we obtain several applications of this fact.
We prove that the unit sphere of every infinite-dimensional uniformly convex Banach space with modulus of convexity $δ$ contains a $(1+\frac12δ(\frac23))$-separated sequence.
The discrete Cesàro operator $ C $ acts continuously in various classical Banach sequence spaces within $ \mathbb{C}^{\mathbb{N}}.$ For the coordinatewise order, many such sequence spaces $ X $ are also complex Banach lattices (eg. $c_0, \ell^p $ for $ 1 < p \leq \infty , $ and $ ces (p)$ for $ p \in \{ 0 \} \cup ( 1, \infty )).$ In such Banach lattice sequence spaces, $ C $ is always a positive operator. Hence, its order spectrum is well defined within the Banach algebra of all regular operators on $ X .$ The purpose of this note is to show, for every $ X $ belonging to the above list of Bana…
We prove that every multiplier M (bounded operator commuting with the shift operator) on a large class of Banach spaces of sequences on Z is associated to a function essentially bounded by the norm of M on the spectrum of S.
We prove that there exist Banach spaces not containing $\ell_1$, failing the point of continuity property and satisfying that every semi-normalized basic sequence has a boundedly complete basic subsequence. This answers in the negative the problem of the Remark 2 in H. P. Rosenthal. "Boundedly complete weak-Cauchy sequences in Banach spaces with PCP." J. Funct. Anal. 253 (2007) 772-781.
It is shown that any Banach space X of sufficiently large density contains an (infinite) unconditional sequence and a separable quotient. If a density of X is a weakly compact cardinal, then X contains an unconditional sequence of cardinality that is equal to the density of X
Let $X$ be a Banach space. We prove that, for a large class of Banach or quasi-Banach spaces $E$ of $X$-valued sequences, the sets $E-\bigcup _{q\inΓ}\ell_{q}(X)$, where $Γ$ is any subset of $(0,\infty]$, and $E-c_{0}(X)$ contain closed infinite-dimensional subspaces of $E$ (if non-empty, of course). This result is applied in several particular cases and it is also shown that the same technique can be used to improve a result on the existence of spaces formed by norm-attaining linear operators.
It is known in Hilbert space frame theory that a Bessel sequence can be expanded to a frame. Contrary to Hilbert space situation, using a result of Casazza and Christensen, we show that there are Banach spaces and approximate Bessel sequences which can not be expanded to approximate Schauder frames. We characterize Banach spaces in which one can expand approximate Bessel sequences to approximate Schauder frames.
We introduce and study the class of almost limited sets in Banach lattices, that is, sets on which every disjoint weak$^{*}$ null sequence of functionals converges uniformly to zero. It is established that a Banach lattice has order continuous norm if and only if almost limited sets and $L$-weakly compact sets coincide. In particular, in terms of almost Dunford-Pettis operators into $c_{0}$, we give an operator characterization of those $σ$-Dedekind complete Banach lattices whose relatively weakly compact sets are almost limited, that is, for a $σ$-Dedekind Banach lattice $E$, every relatively…
Let $1\le p\le\infty$. A Banach lattice $X$ is said to be $p$-disjointly homogeneous or $(p-DH)$ (resp. restricted $(p-DH)$) if every normalized disjoint sequence in $X$ (resp. every normalized sequence of characteristic functions of disjoint subsets) contains a subsequence equivalent in $X$ to the unit vector basis of $\ell_p$. We revisit $DH$-properties of Orlicz spaces and refine some previous results of this topic, showing that $(p-DH)$-property is not stable in the class of Orlicz spaces and the classes of restricted $(p-DH)$ and $(p-DH)$ Orlicz spaces are different. Moreover, we give a c…
In this survey, we present several results related to characterizing the surjective isometries on Banach sequence spaces. Our survey includes full proofs of these characterizations for the classical spaces as well as more recent results for combinatorial Banach spaces and Tsirelson-type spaces. Along the way, we pose many open problems related to the structure of the group of surjective isometries and characterizations of the group of surjective isometries for various Banach spaces.
We consider the closed subspace of $\ell_\infty$ generated by $c_0$ and the characteristic functions of elements of an uncountable, almost disjoint family $\mathcal A$ of infinite subsets of $\mathbb N$. This Banach space has the form $C_0(K_{\mathcal A})$ for a locally compact Hausdorff space $K_{\mathcal A}$ that is known under many names, such as $Ψ$-space and Isbell--Mrówka space. We construct an uncountable, almost disjoint family ${\mathcal A}$ such that the Banach algebra of all bounded linear operators on $C_0(K_{\mathcal A})$ is as small as possible in the sense that every bounded lin…
A continuous operator $T$ between two Banach lattices $E$ and $F$ is called almost order-weakly compact, whenever for each almost order bounded subset $A$ of $E$, $T(A)$ is a relatively weakly compact subset of $F$. In Theorem 4, we show that the positive operator $T$ from $E$ into Dedekind complete $F$ is almost order-weakly compact if and only if $T(x_n) \xrightarrow{\|.\|}0$ in $F$ for each disjoint almost order bounded sequence $\{x_n\}$ in $E$. In this manuscript, we study some properties of this class of operators and its relationships with others known operators.
We show that the solid hull of every weakly precompact set of a Banach lattice $E$ is weakly precompact if and only if every order interval in $E$ is weakly precompact, or equivalently, if and only if every disjoint weakly compact set is weakly precompact. Some results on the domination property for weakly precompact positive operators are obtained. Among other things, we show that, for a pair of Banach lattices $E$ and $F$ with $E$ $σ$-Dedekind complete, every positive operator from $E$ to $F$ dominated by a weakly precompact operator is weakly precompact if and only if either the norm of $E^…
In this paper we study different aspects of the representation of weak*-compact convex sets of the bidual $X^{**}$ of a separable Banach space $X$ via a nested sequence of closed convex bounded sets of $X$.
We show some new results about tilings in Banach spaces. A tiling of a Banach space $X$ is a covering by closed sets with non-empty interior so that they have pairwise disjoint interiors. If moreover the tiles have inner radii uniformly bounded from below, and outer radii uniformly bounded from above, we say that the tiling is normal. In 2010 Preiss constructed a convex normal tiling of the separable Hilbert space. For Banach spaces with Schauder basis we will show that Preiss' result is still true with starshaped tiles instead of convex ones. Also, whenever $X$ is uniformly convex we give pre…
We show that every dominated linear operator from an Banach-Kantorovich space over atomless Dedekind complete vector lattice to a sequence Banach lattice $l_p(Γ)$ or $c_0(Γ)$ is narrow. As a conse- quence, we obtain that an atomless Banach lattice cannot have a finite dimensional decomposition of a certain kind. Finally we show that if a linear dominated operator T from lattice-normed space V to Banach- Kantorovich space W is order narrow then the same is its exact dominant $\ls T\rs$.
In this paper, we introduce (p,q)g-Bessel multipliers in Banach spaces and we show that under some conditions a (p,q)g-Bessel multiplier is invertible. Also, we show the continuous dependency of (p,q)g-Bessel multipliers on their parameters.
The hierarchy of the block bases of transfinite normalized averages of a normalized Schauder basic sequence is introduced and a criterion is given for a normalized weakly null sequence in C(K), the Banach space of scalar valued functions continuous on the compact metric space K, to admit a block basis of normalized averages equivalent to the unit vector basis of c_0, the Banach space of null scalar sequences. As an application of this criterion, it is shown that every normalized weakly null sequence in C(K), for countable K, admits a block basis of normalized averages equivalent to the unit ve…
Let $(x_n)$ be a sequence in a Banach space $X$ which does not converge in norm, and let $E$ be an isomorphically precisely norming set for $X$ such that \[ \sum_n |x^*(x_{n+1}-x_n)|< \infty, \; \forall x^* \in E. \qquad (*) \] Then there exists a subsequence of $(x_n)$ which spans an isomorphically polyhedral Banach space. It follows immediately from results of V. Fonf that the converse is also true: If a separable Banach space $Y$ is a separable isomorphically polyhedral then there exists a non norm convergent sequence $(x_n)$ which spans $Y$ and there exists an isomorphically precisely norm…
In this paper we study the structure of the set $\mbox{Hom}(X,\mathbb{R})$ of all lattice homomorphisms from a Banach lattice $X$ into $\mathbb{R}$. Using the relation among lattice homomorphisms and disjoint families, we prove that the topological dual of the free Banach lattice $FBL(A)$ generated by a set $A$ contains a disjoint family of cardinality $2^{|A|}$, answering a question of B. de Pagter and A.W. Wickstead. We also deal with norm-attaining lattice homomorphisms. For classical Banach lattices, as $c_0$, $L_p$-, and $C(K)$-spaces, every lattice homomorphism on it attains its norm, wh…
We prove new inequalities for the spectral radius, essential spectral radius, operator norm, measure of noncompactness and numerical radius of Hadamard weighted geometric means of positive kernel operators on Banach function and sequence spaces. Several inequalities appear to be new even in the finite dimensional case.
We study the generic behavior of the method of successive approximations for set-valued mappings in Banach spaces. We consider, in particular, the case of those set-valued mappings which are defined by pairs of nonexpansive mappings and give a positive answer to a question raised by Francesco S. de Blasi.
