Multiplication Operators and Dynamical Systems

Let X be a completely regular Hausdorff space, let V be a system of weights on X and let E be a locally convex Hausdorff space. Let CV 0 ( X, E ) and CV b ( X, E ) be the weighted locally convex spaces of vector‐valued continuous functions with a topology generated by seminorms which are weighted analogue of the supremum norm. In the present paper, we characterize multiplication operators and weighted composition operators on the spaces CV 0 ( X, E ) and CV b ( X, E ) induced by scalar‐valued and vector‐valued mappings. A (linear) dynamical system on these weighted spaces is obtained as an application of the theory of multiplication operators.

Extension of operators on spaces of vector-valued continuous functions

A theorem due to Milutin [12] (see also [13]) asserts that for any two uncountable compact metric spaces Ω1 and Ω2 the spaces of continuous real-valued functions C(Ω1) and C(Ω2) are linearly isomorphic. It immediately follows from consideration of tensor products that if X is any Banach space then C(Ω1;X) and C(Ω2;X) are isomorphic.

Topologies β 0 , β 1 , β , β ∞ , β ∞ c {\beta _0},{\beta _1},\beta ,{\beta _\infty },{\beta _{\infty c}} are defined on C b ( X , E ) {C_b}(X,E) , the space of all bounded, continuous functions from a completely regular Hausdorff space X, into E, a normed space, and their duals are determined. Also many properties of these topologies are proved.

A. Pietsch has shown that the class of dominated linear operators on C(5) coincides with the class of absolutely summing operators.If the space C(S) is replaced by CX(S), where A" is a Banach space, this is no longer the case.However, any absolutely summing operator is always dominated, and the classes of operators coincide exactly when X is finite dimensional.A characterization of absolutely summing operators on CX(S) is given.

Local symmetry and smoothness in the space of vector-valued continuous functions

A class of nonlinear variational inequalities involving strongly G-monotone mappings in a locally convex Hausdorff topological vector space setting is presented. The obtained result generalizes the results on nonlinear variational inequalities on monotone and strongly monotone mappings.

Altman [1] showed that Riesz-Schauder theory remains valid for a completely continuous linear operator on a locally convex Hausdorflf topological vector space over the complex field. In a later paper [2], he proved an analogue of the Aronszajn-Smith result; specifically, he showed that such an operator possesses a proper closed invariant subspace. The purpose of this paper is to show that Ringrose's theory of superdiagonal forms for compact linear operators [3] can be generalized to the case of a completely continuous linear operator on a locally convex Hausdorff topological vector space over the complex field. However, the proof given in [3] requires considerable modification.

In this note an analytic representation is given for continuous linear operators from C ( X ) C(X) into a linear normed space Y where C ( X ) C(X) is the space of continuous functions on [0, 1] with values in a linear normed space X.

Recently Huang et al. (Math. Comput. Model. 43:1267-1274, 2006)introduced a class of parametric implicit vector equilibrium problems (for shortPIVEP) and they presented some existence results for a solution of PIVEP. Also,they provided two theorems about upper and lower semi-continuity of the solutionset of PIVEP in a locally convex Hausdorff topological vector space. The paperextends the corresponding results obtained in the setting of topological vectorspaces with mild assumptions and removing the notion of locally non-positivenessat a point and lower semi-continuity of the parametric mapping.

In [1] it is shown that barrelledness and quasi-barrelledness are merely the two extreme examples of a property, called 2-symmetry, which may be possessed by a locally convex Hausdorff topological vector space. The object of this note is to show how recent characterisations [2; 3] of barrelled and quasi-barrelled spaces may be subsumed under characterisations of I-symmetric spaces, and to exhibit some properties of these spaces. First we need some definitions and simple results.

This chapter is about metric spaces, an abstract generalization of the real line that allows discussion of open and closed sets, limits, convergence, continuity, and similar properties. The usual distance function for the real line becomes an example of a metric. The other notions are defined in terms of the metric. The advantage of the generalization is that proofs of certain properties of the real line immediately go over to all other examples. Section 1 gives the definition of metric space and open set, and it lists a number of important examples, including Euclidean spaces and certain spaces of functions. Sections 2 through 4 develop properties of open and closed sets, continuity, and convergence of sequences that are simple generalizations of known facts about $\mathbb{R}$. Section 5 shows how a subset of a metric space can be made into a metric space so that the restriction of a continuous function from the whole space to the subset remains continuous. It also shows that three natural metrics for the product of two metric spaces lead to the same open sets, continuous functions, and convergent sequences. Section 6 shows that any metric space is “Hausdorff,” “regular,” and “normal,” and it goes on to exhibit three different countability hypotheses about a metric space as equivalent. A metric space with these properties is called “separable.” Section 7 concerns compactness and completeness. A metric space is defined to be “compact” if every open cover has a finite subcover. This property is equivalent to the condition that every sequence has a convergent subsequence. The Heine–Borel Theorem says that the compact sets of $\mathbb{R}^n$ are exactly the closed bounded sets. A number of the results early in Chapter I that were proved by the Bolzano–Weierstrass Theorem in the context of the real line are seen to extend to any compact metric space. A metric space is “complete” if every Cauchy sequence is convergent. A metric space is compact if and only if it is complete and “totally bounded.” Section 8 concerns connectedness, which is an abstraction of the property of an interval of the line that accounts for the Intermediate Value Theorem. Section 9 proves a fundamental result known as the Baire Category Theorem. A sample consequence of the theorem is that the pointwise limit of a sequence of continuous complex-valued functions on a complete metric space must have points where it is continuous. Section 10 studies the spaces of real-valued and complex-valued continuous functions on a compact metric space. A generalization of Ascoli's Theorem from the setting of Chapter I provides a characterization of compact sets in either of these spaces of continuous functions. A generalization of the Weierstrass Approximation Theorem, known as the Stone–Weierstrass Theorem, gives sufficient conditions for a subalgebra of either of these spaces of continuous functions to be dense. One consequence is that these spaces of continuous functions are separable. Section 11 constructs the “completion” of a metric space out of Cauchy sequences in the given space. The result is a complete metric space and a distance-preserving map of the given metric space into the completion such that the image is dense.

LetVbe an arbitrary system of weights on an open connected subsetGofℂN(N≥1)and letB(E)be the Banach algebra of all bounded linear operators on a Banach spaceE. LetHVb(G,E)andHV0(G,E)be the weighted locally convex spaces of vector-valued analytic functions. In this survey, we present a development of the theory of multiplication operators and composition operators from classical spaces of analytic functionsH(G)to the weighted spaces of analytic functionsHVb(G,E)andHV0(G,E).

Asymptotic closure condition and Fenchel duality for DC optimization problems in locally convex spaces

Given a Banach lattice L, the space of lattice Lipschitz operators on L has been introduced as a natural Lipschitz generalization of the linear notions of diagonal operator and multiplication operator on Banach function lattices. It is a particular space of superposition operators on Banach lattices. Motivated by certain procedures in Reinforcement Learning based on McShane–Whitney extensions of Lipschitz maps, this class has proven to be useful also in the classical context of Mathematical Analysis. In this paper, we discuss the properties of such operators when defined on spaces of continuous functions, focusing attention on the functional bounds for the pointwise Lipschitz inequalities defining the lattice Lipschitz operators, the representation theorems for these operators as vector-valued functions, and the corresponding dual spaces. Finally, and with possible applications in Artificial Intelligence in mind, we provide a McShane–Whitney extension theorem for these operators.