Covering properties on X2⧹Δ, W-sets,and compact subsets of Σ-products

Let Δ ? X¹ be the diagonal. In the first part of this paper, we show that a compact space X is Corson compact (resp., Eberlein compact; compact metric) if and only if X²?Δ is metalindelöf (resp., σ-metacompact; paracompact). In the second part of the paper, we investigate the notion of a W-set in a space X, which is defined in terms of an infinite game. We show that a compact space X is Corson compact if and only if X has a W-set diagonal, and that a compact scattered space X is strong Eberlein compact if and only if each point of X is a W-set in X.     

Cut points in some connected topological spaces

We prove that a connected topological space with endpoints has exactly two non-cut points and every cut point is a strong cut point; it follows that such a space is a COTS and the only two non-cut points turn out to be endpoints (in each of the two orders) of the COTS. A non-indiscrete connected topological space with exactly two non-cut points and having only finitely many closed points is proved homeomorphic to a finite subspace of the Khalimsky line. Further, it is shown, without assuming any separation axiom, that in a connected and locally connected topological space X, for a, b in X, S[a,b] is compact whenever it is closed. Using this result we show that an H(i) connected and locally connected topological space with exactly two non-cut points is a compact COTS with end points.     

On a calculus for a generalized scalar operator

Let $\text{X}$ be a Banach space; S and T bounded scalar-type operators in $\text{X}$. Define Δ on the space of bounded operators on $\text{X}$ by ΔX = TX ? XS if X is a bounded operator. We set up a calculus for Δ which allows us to consider f(Δ), for f a complex-valued bounded Borel measurable function on the spectrum of Δ, as an operator in the space of bounded operators whose domain is a subspace of operators which we call measure generating. This calculus is used to obtain some results on when the kernel of Δ is a complemented subspace of the space of bounded operators on $\text{X}$.     

Spectral reciprocity and matrix representations of unbounded operators

We study a family of unbounded Hermitian operators in Hilbert space which generalize the usual graph-theoretic discrete Laplacian. For an infinite discrete set X, we consider operators acting on Hilbert spaces of functions on X, and their representations as infinite matrices; the focus is on ?²(X), and the energy space HE. In particular, we prove that these operators are always essentially self-adjoint on ?²(X), but may fail to be essentially self-adjoint on HE. In the general case, we examine the von Neumann deficiency indices of these operators and explore their relevance in mathematical physics. Finally we study the spectra of the HE operators with the use of a new approximation scheme.     

A generalisation of the functional approach to compactness

A topological space X is compact iff the projection π:X×Y→Y is closed for any space Y. Taking this as a definition and then asking that π maps α-closed subspaces of X×Y onto β-closed subspaces of Y, for different closures α and β, extends the notion of compactness to include also examples of “asymmetric compactness” pursued in the article.Categorical closure operators and a so-called “functional approach to general topology” are employed to define and prove fundamental properties of compact objects and proper maps in this generalised setting.     

Lipschitz compact operators

We introduce the notion of Lipschitz compact (weakly compact, finite-rank, approximable) operators from a pointed metric space X into a Banach space E. We prove that every strongly Lipschitz p-nuclear operator is Lipschitz compact and every strongly Lipschitz p-integral operator is Lipschitz weakly compact. A theory of Lipschitz compact (weakly compact, finite-rank) operators which closely parallels the theory for linear operators is developed. In terms of the Lipschitz transpose map of a Lipschitz operator, we state Lipschitz versions of Schauder type theorems on the (weak) compactness of the adjoint of a (weakly) compact linear operator.     

Vector measures and strict topologies

Let X be a completely regular Hausdorff space and Cb(X) be the space of all real-valued bounded continuous functions on X, endowed with the strict topology βσ. We study topological properties of continuous and weakly compact operators from Cb(X) to a locally convex Hausdorff space in terms of their representing vector measures. In particular, Alexandrov representation type theorems are derived. Moreover, a Yosida-Hewitt type decomposition for weakly compact operators on Cb(X) is given.     

The determinant bundle on the moduli space of stable triples over a curve

We construct a holomorphic Hermitian line bundle over the moduli space of stable triples of the form (E₁, E₂,?), where E₁ and E₂ are holomorphic vector bundles over a fixed compact Riemann surfaceX, and?: E₂ → E₁ is a holomorphic vector bundle homomorphism. The curvature of the Chern connection of this holomorphic Hermitian line bundle is computed. The curvature is shown to coincide with a constant scalar multiple of the natural Kähler form on the moduli space. The construction is based on a result of Quillen on the determinant line bundle over the space of Dolbeault operators on a fixed C^∞ Hermitian vector bundle over a compact Riemann surface.     

The Choquet integral representability of comonotonically additive functionals in locally compact spaces

We give an alternative and direct approach to the Choquet integral representability of a comonotonically additive, bounded, monotone functional I defined on the space of all continuous, real-valued functions on a locally compact space X with compact support and on the space of all continuous, real-valued functions on X vanishing at infinity. To this end, we introduce the notion of the asymptotic translatability of the functional I and show that this simple notion is equivalent to the Choquet integral representability of I with respect to a monotone measure on X with appropriate regularity.     

Surjective isometries on spaces of vector valued continuous and Lipschitz functions

For X a compact Hausdorff topological space and F a strictly convex and reflexive Banach space we characterize the surjective isometries of certain subspaces of C(X, F). It follows from this characterization that surjective isometries on spaces of vector valued Lipschitz functions equipped with a p-norm are also weighted composition operators.     

Non-regular power homogeneous spaces

We show that the cardinality of any space X with Δ-power homogeneous semiregularization that is either Urysohn or quasiregular is bounded by ²c(X)πχ(X). This improves a result of G.J. Ridderbos who showed this bound holds for Δ-power homogeneous regular spaces. By introducing the notion of a local πθ-base, we show that this bound can be further sharpened. We also show that no H-closed extremally disconnected space is power homogeneous. This is a variation of a result of K. Kunen who showed that no compact F-space is power homogeneous.     

Injective dimension of sheaves of rational vector spaces

The Cantor–Bendixson rank of a topological space X is a measure of the complexity of the topology of X. We will be interested primarily in the case that the space is profinite: Hausdorff, compact and totally disconnected. In this paper, we prove that the injective dimension of the abelian category of sheaves of $\mathbb{Q}$-modules over a profinite space X is determined by the Cantor–Bendixson rank of X.     

On the product of homogeneous spaces

Within the class of Tychonoff spaces, and within the class of topological groups, most of the natural questions concerning ‘productive closure’ of the subclasses of countably compact and pseudocompact spaces are answered by the following three well-known results: (1) [ZFC] There is a countably compact Tychonoff space X such that X × X is not pseudocompact; (2) [ZFC] The product of any set of pseudocompact topological groups is pseudocompact; and (3) [ZFC+ MA] There are countably compact topological groups G₀, G₁ such that G₀ × G₁ is not countably compact.In this paper we consider the question of ‘productive closure” in the intermediate class of homogeneous spaces. Our principal result, whose proof leans heavily on a simple, elegant result of V.V. Uspenski?, is this: In ZFC there are pseudocompact, homogeneous spaces X₀, X₁ such that X₀ × X₁ is not pseudocompact; if in addition MA is assumed, the spaces X_i may be chosen countably compact.Our construction yields an unexpected corollary in a different direction: Every compact space embeds as a retract in a countably compact, homogeneous space. Thus for every cardinal number α there is a countably compact, homogeneous space whose Souslin number exceeds α.     

Topological-algebraic properties of function spaces with set-open topologies

For a Tychonoff space X, we denote by Cλ(X) the space of all real-valued continuous functions on X with set-open topology. In this paper, we study the topological-algebraic properties of Cλ(X). Our main results state that (1) Cλ(X) is a topological vector space (a topological group) iff λ is a family of C-compact sets and Cλ(X)=Cλ^′(X), where λ^′ consists of all C-compact subsets of every set of λ. In particular, if Cλ(X) is a topological group, then the set-open topology coincides with the topology of uniform convergence on a family λ; (2) a topological group Cλ(X) is ω-narrow iff λ is a family of metrizable compact subsets of X.     

Universal spaces of subdifferentials of sublinear operators ranging in the cone of bounded lower semicontinuous functions

We study Fréchet’s problem of the universal space for the subdifferentials ?P of continuous sublinear operators P: V → BC(X)_～ which are defined on separable Banach spaces V and range in the cone BC(X)_～ of bounded lower semicontinuous functions on a normal topological space X. We prove that the space of linear compact operators L ^c(? ², C(βX)) is universal in the topology of simple convergence. Here ? ² is a separable Hilbert space, and βX is the Stone-?ech compactification of X. We show that the images of subdifferentials are also subdifferentials of sublinear operators.     

Almost compactness in fuzzy topological spaces

We introduce and study almost compactness for fuzzy topological spaces. We show that the almost continuous image of an almost compact fuzzy topological space is almost compact. Moreover, we show that generally almost compactness for fuzzy topological spaces is not product-invariant, but if X and Y are almost fuzzy topological spaces and X is product related to Y, then their fuzzy topological product is almost compact.     

On a sequence of integral operators on weighted L p spaces

Continuing some investigations started in previous papers, we introduce and study a sequence of multidimensional positive integral operators which generalize the Gauss-Weierstrass operators. We show that this sequence is an approximation process in some classes of weighted L ^p spaces on ? ^N , N ≥ 1. Estimates of the rate of convergence are also obtained. Our mean tool is a Korovkin-type theorem which we establish in the context of L ^p (X, µ) spaces, X being a locally compact Hausdorff space and µ a regular positive Borel measure on X. Several examples are explicitly indicated as well.     

Riesz* homomorphisms on pre-Riesz spaces consisting of continuous functions

In the theory of operators on a Riesz space (vector lattice), an important result states that the Riesz homomorphisms (lattice homomorphisms) on C(X) are exactly the weighted composition operators. We extend this result to Riesz* homomorphisms on order dense subspaces of C(X). On those subspace we consider and compare various classes of operators that extend the notion of a Riesz homomorphism. Furthermore, using the weighted composition structure of Riesz* homomorphisms we obtain several results concerning bijective Riesz* homomorphisms. In particular, we characterize the automorphism group for order dense subspaces of C(X). Lastly, we develop a similar theory for Riesz* homomorphisms on subspace of \(C_0(X)\), for a locally compact Hausdorff space X, and apply it to smooth manifolds and Sobolev spaces.     

Hereditary normality-type properties of hyperspaces

Let X be a Hausdorff topological space and exp(X) be the space of all (nonempty) closed subsets of a space X with the Vietoris topology. We consider hereditary normality-type properties of exp(X). In particular, we prove that if exp(X) is hereditarily D-normal, then X is a metrizable compact space.     

New criterion of the approximation property

This paper is concerned with the approximation property which is an important property in Banach space theory. We show that a Banach space X has the approximation property if (and only if), for every Banach space Y, the set of finite rank operators from X to Y is dense in the corresponding space of compact operators, in the usual topology of uniform convergence on compact sets.