Order Continuous Operators on Pre-Riesz Spaces and Embeddings

If the vector space \({\mathcal{L}}^r(E, F)\) of all regular operators between the vector lattices E and F is ordered by the collection of its positive operators, then the Dedekind completeness of F is a sufficient condition for \({\mathcal{L}}^r(E, F)\) to be a vector lattice. \({\mathcal{L}}^r(E, F)\) and some of its subspaces might be vector lattices also in a more general situation. In the paper we deal with ordered vector spaces \({\mathcal{V}}(E, F)\) of linear operators and ask under which conditions are they vector lattices, lattice-subspaces of the ordered vector space \({\mathcal{L}}^r(E, F)\) or, in the case that \({\mathcal{L}}^r(E, F)\) is a vector lattice, sublattices or even Banach lattices when equipped with the regular norm. The answer is affirmative for many classes of operators such as compact, weakly compact, regular AM-compact, regular Dunford-Pettis operators and others if acting between appropriate Banach lattices. Then it is possible to study the finite elements in such vector lattices \({\mathcal{V}}(E, F)\), where F is not necessary Dedekind complete. In the last part of the paper there will be considered the question how the order structures of E, F and \({\mathcal{V}}(E, F)\) are mutually related. It is also shown that those rank one and finite rank operators, which are constructed by means of finite elements from E′ and F, are finite elements in \({\mathcal{V}}(E,F)\). The paper contains also some generalization of results obtained for the case \({\mathcal{V}}(E, F) = {\mathcal{L}}^r(E, F)\) in [10].

Riesz homomorphisms between vector lattices are characterized by interval preserving order adjoints. Buskes, van Rooij, and van Haandel generalized the notion of a Riesz homomorphism to linear maps between general ordered vector spaces. In this paper, we discuss how far the characterization between vector lattices extends to directed ordered vector spaces and pre-Riesz spaces.

Problems of solving different linear difference equation is given to study the properties of the matrix operators in various functional spaces. One of the important problems of functional analysis is to establish criteria of boundedness of the linear operators in functional spaces. Question of the boundedness of matrix operators in sequence spaces is a classic problem of functional analysis and there are many unsolved problems in it. For example, in the general case it is impossible to establish the boundedness of the matrix operator in the spaces of sequences by the given matrix. Therefore, various classes of matrix operators are considered for which the criteria of their boundedness are known. Due to the variety of encountered problems in practice, it is necessary to have various alternative criteria for the boundedness of matrix operators. In this paper, we establish a new alternative criterion for the boundedness of one class of matrix operators.

The order continuous operators consider one of important topic in functional analysis and its applications, the affiliations among order continuous operators and the other classes of operators such as -order are continuous, order bounded, and singular operators, have been studied and investigated, we proved that if an order bounded operator concerning two Riesz space with Dedekind complete is continuous and ordered, then is order continuous, and this paper shows that if is space that is countable, now is not -order continuous, while is uncountable, then is necessarily -order continuous, by giving an example we showed that null ideal for the operator is band when is bounded ordered, further, it is ordered and continuous. Finally, we concluded the operator that is a positively and orderly continuous map on ordered dense with memorizing Riesz subspace of a Riesz space with its range is Dedekind complete, it has only unique ordered continuous expansion all of space.

We study conformal conserved currents in arbitrary irreducible representations of the Lorentz group using the embedding space formalism. With the help of the operator product expansion, we first show that conservation conditions can be fully investigated by considering only two- and three-point correlation functions. We then find an explicitly conformally-covariant differential operator in embedding space that implements conservation based on the standard position space operator product expansion differential operator ∂μ, although the latter does not uplift to embedding space covariantly. The differential operator in embedding space that imposes conservation is the same differential operator mathcal{D} ijA used in the operator product expansion in embedding space. We provide several examples including conserved currents in irreducible representations that are not symmetric and traceless. With an eye on four-point conformal bootstrap equations for four conserved vector currents 〈JJJJ〉 and four energy-momentum tensors 〈TTTT〉, we mostly focus on conservation conditions for leftlangle JJmathcal{O}rightrangle and leftlangle TTmathcal{O}rightrangle . Finally, we reproduce and extend the consequences of conformal Ward identities at coincident points by determining three-point coefficients in terms of charges.

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.

We develop techniques useful for obtaining conformal blocks in embedding space. We construct a unique differential operator in embedding space and use it to construct a function that will be an important ingredient in assembling conformal blocks. We show a number of relations that the components of conformal blocks satisfy and find invariance of our expressions under the dihedral group.

This work studies boundedness properties of the fractional maximal operator on metric measure spaces under standard assumptions on the measure. The main motivation is to show that the fractional maximal operator has similar smoothing and mapping properties as the Riesz potential. Instead of the usual fractional maximal operator, we also consider a so-called discrete maximal operator which has better regularity. We study the boundedness of the discrete fractional maximal operator in Sobolev, Holder, Morrey, and Campanato spaces. We also prove a version of the Coifman– Rochberg lemma for the fractional maximal function.

Chapter 2 Positive operators

The spectral properties of a compact operator \(T : X \longrightarrow Y\) on a normed linear space resemble those of square matrices. For a compact operator, the spectral properties can be treated fairly completely in the sense that Fredholm's famous theory of integral equations may be extended to linear functional equations \(T x -\lambda\) \(= y\) with a complex parameter \(\lambda\) . This paper has studied and investigated the spectral properties of compact operators in Hilbert spaces. The spectral properties of compact linear operators are relatively simple generalization of the eigenvalues of finite matrices. As a result, the paper has given a number of corresponding propositions and interesting facts which are used to prove basic properties of compact operators. The Fredholm theory has been introduced to investigate the solvability of linear integral equations involving compact operators.

In this chapter we are concerned with the basic properties of the superposition operator in so-called ideal spaces which are, roughly speaking, Banach spaces of measurable functions with monotone norm. To formulate our results in a sufficiently general framework, we must introduce a large number of auxiliary notions which will be justified by the results in concrete function spaces given in subsequent chapters; we request the reader's indulgence until then. First, we give conditions for the local and global boundedness of the superposition operator F between ideal spaces X and Y which are typically ensured by special properties of the “source space” X . Second, special properties, such as absolute boundedness and compactness, are treated. Afterwards, we give conditions for the continuity and uniform continuity of F which are now typically ensured by special properties of the “target space” Y . For example, F is “always” continuous if Y is regular, and “never” continuous if Y is completely irregular (see the definitions below). Weak continuity of F between ideal spaces is also considered; here we mention the surprising fact that, loosely speaking, only linear superposition operators are weakly continuous. Next, we give necessary and sufficient conditions under which F satisfies a Lipschitz or Darbo condition. It turns out that in many spaces these two conditions are in fact equivalent. Finally, the last part of this chapter is concerned with differentiability conditions for the superposition operator between ideal spaces.

H-Accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces

We study orthogonal and symmetric operators in non-Archimedean Hilbert spaces in the connection with p-adic quantization. This quantization describes measurements with finite precision. Symmetric (bounded) operators in the p-adic Hilbert spaces represent physical observables. We study spectral properties of one of the most important quantum operators, namely, the operator of the position (which is represented in the p-adic Hilbert L2-space with respect to the p-adic Gaussian measure). Orthogonal isometric isomorphisms of p-adic Hilbert spaces preserve precisions of measurements. We study properties of orthogonal operators. It is proved that each orthogonal operator in the non-Archimedean Hilbert space is continuous. However, there exist discontinuous operators with the dense domain of definition which preserve the inner product. There also exist nonisometric orthogonal operators. We describe some classes of orthogonal isometric operators and we study some general questions of the theory of non-Archimedean Hilbert spaces (in particular, general connections between topology, norm and inner product).

The purpose of this note is to present certain aspects of the theory of spectral operators in Grothendieck spaces with the Dunford-Pettis property, briefly, GDP-spaces, thereby elaborating on the recent note [10].For example, the sum and product of commuting spectral operators in such spaces are again spectral operators (cf. Proposition 2.1) and a continuous linear operator is spectral if and only if it has finite spectrum (cf. Proposition 2.2). Accordingly, if a spectral operator is of finite type, then its spectrum consists entirely of eigenvalues. Furthermore, it turns out that there are no unbounded spectral operators in such spaces (cf. Proposition 2.4). As a simple application of these results we are able to determine which multiplication operators in certain function spaces are spectral operators.

One of the principal results of the paper is that each scalar-type spectral operator in the quasicomplete locally convex space X is reflexive.The paper also studies in detail the relation between the theory of equicontinuous spectral measures in locally convex spaces and the order properties of equicontinuous Bade complete Boolean algebras of projections.o.Introduction.One of the principal results of this paper is that each scalar-type spectral operator T in the quasicomplete, locally convex space X is reflexive, i.e. the strongly closed subalgebra generated by the identity and T in .P(X), the space of continuous linear operators on X, consists precisely of those continuous linear operators on X which leave invariant each (closed) T-invariant subspace of X.For the case that X is a Banach space, this result was established by Gillespie [9] via an interesting factorization theorem in Banach function spaces, a method which does not appear to extend readily to the more general setting.The present approach, however, avoids factorization theorems by showing directly that each continuous linear functional on the strongly closed algebra generated by a Bade-complete, equicontinuous Boolean algebra of projections in X has a representation of the form < .x, x') for some x E X and x' E X', where X' denotes the dual space of X, a result which goes back to R. Pallu de la Barriere [20] for the case of Abelian von Neumann algebras in Hilbert space.Our method is based on ideas from the theory of Riesz spaces and yields considerable simplification of technique, even in the setting of Banach spaces.The cornerstone of the present paper is the extension of the reflexivity theorem of Bade [2] to the setting of locally convex spaces proved in [6] via the theory of closed spectral measures and further refined and sharpened in [5] using purely intrinsic methods, based on order considerations.One of the new features which emerged from the approach of [5] was a type of "automatic continuity" theorem for a certain class of everywhere defined linear operators, even in the absence of a suitable closed-graph theorem.This idea is exploited in 1 to show that an everywhere