Convergence structures and locally solid topologies on vector lattices of operators

Let L be a vector lattice, "(" x_α ") " be a L-valued net, and x∈L . If |x_α-x|∧u→┴o 0 for every u ∈〖 L〗_+ then it is said that the net "(" x_α ")" unbounded order converges to x and is denoted by □(x_α □(→┴uo x)) . This definition of unbounded order convergence has been extensively studied on many structures, including vector lattices, local solid vector lattices, normed lattices and lattice normed spaces. It is not possible to apply this type of convergence to general lattices due to the lack of algebraic structure. Therefore, we will use a type of convergence that is considered to be the motivation for this type of convergence, first defined as independent convergence in semi-ordered linear spaces and later called unbounded order convergence. Namely, L is a lattice, x_α is an L -valued net, and x ϵ L . If (x_α∧b )∨a order converges to (x∧b )∨a for every a,b∈L with a≤b, then it is said that "(" x_α ")" individual converges to x or unbounded order converges to x . This definition can be easily applied to general lattices. In this article, this definition will be understood as unbounded order convergence. Also, even if these two convergences are called by the same name, there is no equivalence between them for general lattices, an example of this is mentioned in this article. Let L be a partially ordered set, "(" x_α ")" be an L -valued net and x∈L (x_α) is said to be star convergent to x if every subnet of the net (x_α ) has a subnet that is order convergent to x and denoted by x_α □(→┴s x). In this paper, a new type of convergence on lattices is defined by combining unbounded order convergence (individual convergence) and star convergence. Let L be a lattice, (x_α ) a net and x∈L (x_α) is said to be unbounded star convergent to x if for every subnet (x_β) of (x_α), there exists a subnet (x_ζ) of (x_β) such that (x_ζ∧b)∨ □(a→┴o ) (x∧b)∨a for every a,b∈L with a≤b and it is denoted by x_α □(→┴us x). The differences between the new type of convergence, called unbounded star convergence, and order convergence, star convergence are demonstrated with counterexamples. The meaningfulness of the unbounded star convergence type is analyzed with these counterexamples and the implications presented. In addition, basic questions about unbounded star convergence of a given net on lattices such as convergence of a fixed net, uniqueness of the limit, convergence of the subnet of a convergent net are answered.

Unbounded order convergence in dual spaces

In this paper, we continue investigation of the lateral order on vector lattices started in [25]. We consider the complexification of a real vector lattice E and introduce the lateral order on . Our first main result asserts that the set of all fragments of an element of the complexification of an uniformly complete vector lattice E is a Boolean algebra. Then, we study narrow operators defined on the complexification of a vector lattice E, extending the results of articles [22, 27, 28] to the setting of operators defined on complex vector lattices. We prove that every order‐to‐norm continuous linear operator from the complexification of an atomless Dedekind complete vector lattice E to a finite‐dimensional Banach space X is strictly narrow. Then, we prove that every C‐compact order‐to‐norm continuous linear operator from to a Banach space X is narrow. We also show that every regular order‐no‐norm continuous linear operator from to a complex Banach lattice is narrow. Finally, in the last part of the paper we investigate narrow operators taking values in symmetric ideals of compact operators.

A classical result due to M. Eidelheit and B. Yood states that the standard algebra norm on the algebra of bounded linear operators on a Banach space is minimal, in the sense that the norm must be less than a multiple of any other submultiplicative norm on the same algebra. This definition does not assume that the arbitrary algebra norm is complete. In cases when the standard algebra norm is, in addition, maximal, it is therefore unique up to equivalence. More recently, M. Meyer showed that the Calkin algebras of a very restricted class of Banach spaces also have unique algebra norms. We generalise the Eidelheit-Yood method of proof, to show that the conventional quotient norm on a larger class of Calkin algebras is minimal. Since maximality of the norm is a presumed property for the class, the norm is also unique. We thus extend the result of Meyer. In particular, we establish that the Calkin algebras of canonical Banach spaces such as James’ space and Tsirelson’s space have unique algebra norms, without assuming completeness. We also prove uniqueness of norm for quotients of the algebras of operators on classical non-separable spaces, the closed ideals of which were previously studied by M. Daws. One aspect of the Eidelheit-Yood method is a dependence on the uniform boundedness principle. As a component of our generalisation, we prove an analogue of that principle which applies to Calkin algebra elements rather than bounded linear operators. In order to translate the uniform boundedness principle into this new setting, we take the perspective that non-compact operators map certain wellseparated sequences to other well-separated sequences. We analyse the limiting separation of such sequences, using these values to measure the non-compactness of operators and define the requisite notion of a bounded set of non-compact operators. In the cases when the underlying Banach space has a Schauder basis, we are able to restrict attention to seminormalised block basic sequences. As a consequence, our main uniqueness of norm result for Calkin algebras relies on the existence of bounded mappings between, and projections onto, the spans of block basic sequences in the relevant Banach spaces.

On C-compact orthogonally additive operators

In a previous paper we defined a Denjoy integral for mappings from a vector lattice to a complete vector lattice. In this paper we define a Henstock-Kurzweil integral for mappings from a vector lattice to a complete vector lattice and consider the relation between these two integrals.

1. Introduction. A linear operator T on a vector lattice is band preserving if x⊥ y implies Tx ⊥ y. Much is known about the order bounded band preserving operators on an Archimedean vector lattice. The collection of all of these forms an Abelian algebra under composition and a vector lattice for the operator order (see [7], [8] and [13] amongst others). Very little appears to be known about band preserving operators which are not order bounded apart from some isolated examples ([11], [13], [1] and [17]) and some non-existence results ([11] and [1]).

Fremlin tensor product behaves well with the unbounded order convergence

In order to find an extension of Brezis–Lieb’s lemma to the case of nets, we replace the almost everywhere convergence by the unbounded order convergence and introduce the pre-Brezis–Lieb property in normed lattices. Then we identify a wide class of Banach lattices in which the Brezis–Lieb lemma holds true. Among other things, it gives an extension of the Brezis–Lieb lemma for nets in \(L^p\) for \(p\in [1,\infty )\).

Chapter 2 Positive operators

Several recent papers investigated unbounded convergences in Banach lattices. The focus of this paper is to apply the results of unbounded convergence to the classical Banach lattice theory from a new perspective. Combining all unbounded convergences, including unbounded order (norm, absolute weak, absolute weak*) convergence, we characterize L-weakly compact sets, L-weakly compact operators and M-weakly compact operators on Banach lattices. For applications, we introduce so-called statistical-unbounded convergence and use these convergences to describe KB-spaces and reflexive Banach lattices.

Setting forth the problems Some history Synopsis of the main results Preliminaries The McPolin-Wickstead and Huijsmans-de Pagter-Koldunov Theorems revisited d-bases Band preserving operators and band-projections Central operators and Problems A and B Range-domain exchange in the Huijsmans-de Pagter-Koldunov Theorem d-splitting number of disjointness preserving operators Essentially one-dimensional and discrete vector lattices Essentially constant functions and operators on $C[0,1]$ Counterexamples Dedekind complete vector lattices and Problems A and B Generalizations to $(r_u)$-complete vector lattices Open problems References Index.

It is proved that an almost f-algebra multiplication and a d-algebra multiplication defined on a majorizing vector sublattice of a Dedekind complete vector lattice can be extended to the whole vector lattice by using purely algebraic and order theoretical means.

A variation on the uniform boundedness theorem [2, p. 105] of functional analysis is obtained by replacing pointwise boundedness by a slightly stronger type of boundedness. This makes it possible to remove the category requirements from the set over which it is desired to have a collection of functions uniformly bounded. The concept of boundedness to be utilized is called boundedness on Cauchy sequences (see Definition 1). When a Cauchy sequence converges in the domain of a family of continuous functions, boundedness on a Cauchy sequence is equivalent to pointwise boundedness of the family at the limit point of the point of the sequence. When the domain of the continuous functions is a metric space, boundedness on the collection on all Cauchy sequences is equivalent to pointwise boundedness of the continuous extensions on the completion of the domain. The interesting case is when the domain is not metrizable because boundedness on Cauchy sequences does not necessarily imply pointwise boundedness of the extended functions on the completion or even on the sequential completion of the domain. The main result of the paper is given in Theorem 1 which is followed by six corollaries. The sixth corollary is especially interesting in stating that the closure of an infrabarrelled space in its second adjoint space is a barreled space. The theorem on uniform boundedness immediately suggests an absorption theorem which is Theorem 2 of the paper. Theorem 1 does not follow as a corollary of Theorem 2 as is the case of the classical uniform boundedness theorem [2, p. 105]. DEFINITION 1. A family H of continuous linear transformations with common domain a linear topological space E and ranges in a locally convex space F is bounded on a Cauchy sequence {Xn } in E if for every neighborhood V of the origin in F there exists a positive real number r with the property that for each f in H there is an Nf such that f (xn) is in r V for all n > Nf. The following theorem is stated in the context of the definition.

Let B(X) be the operator algebra for a separable infinite dimensional Hilbert space H, endowed with the strong operator topology or *-strong topology. We give sufficient conditions for a continuous linear mapping L: B(X) → B(X) to be supercyclic or *-supercyclic. In particular our condition implies the existence of an infinite dimensional subspace of supercyclic vectors for a mapping T on H. Hypercyclicity of the operator algebra with strong operator topology was studied by Chan and here we obtain an analogous result in the case of *-strong operator topology.