Let X be a Banach space and Z a nonempty closed subset of X. Let J :Z → R be an upper semicontinuous function bounded from above. This paper is concerned with the perturbed optimization problem supz∈Z{J (z)+ ‖x − z‖}, which is denoted by (x, J )-sup. We shall prove in the present paper that if Z is a closed boundedly relatively weakly compact nonempty subset, then the set of all x ∈X for which ...

An E0-semigroup is a semigroup of unital endomorphisms of B(H), the set of bounded operators on an infinite dimensional separable Hilbert space H , with appropriate continuity. Despite the fact that B(H) is the simplest infinite dimensional factor, the classification of E0-semigroups is far from a satisfactory stage mainly caused by the presence of so called type II and type III examples [11],[...

We show that the space of all Lelek fans in a Cantor fan, equipped with the Hausdorff metric, is homeomorphic to the separable Hilbert space. This result is a special case of a general theorem we prove about spaces of upper semicontinuous functions on compact metric spaces that are strongly discontinuous.

Let X and Y be time-homogeneous Markov processes with common state space E, and assume that the transition kernels of X and Y admit densities with respect to suitable reference measures. We show that if there is a time t > 0 such that, for each x ∈ E, the conditional distribution of (Xs)0≤s≤t, given X0 = x = Xt, coincides with the conditional distribution of (Ys)0≤s≤t, given Y0 = x = Yt, then t...

The present work is a first step towards a systematic constructive development of the theory of operator algebras over a Hilbert space H. Among the topics investigated in the thesis are locally convex topologies, the extension and characterisation of ultraweakly continuous linear functionals on B(H), and conditions that ensure the (constructive) existence of the adjoint of a bounded linear oper...

We prove that the laws of diiusion processes M on E associated with Dirichlet forms of type E(u; v) = R E hA(z)ru(z); rv(z)i H (dz), where H, E are separable Hilbert spaces, are the weak limits of laws of nite dimensional diiusions. These are associated with the image Dirichlet forms obtained from E under projections from E onto nite dimensional subspaces in H. As a by-product we obtain Hoelder...

The purpose of this paper is to study singular integrals whose kernels k(x; ξ) are variable, i.e. they depend on some parameter x ∈ R and in ξ ∈ R \ {0} satisfy mixed homogeneity condition of the form k(x;μξ1, . . . , μ ξn) = μ − ∑ n i=1 ik(x; ξ) with positive real numbers αi ≥ 1 and μ > 0. The continuity of these operators in L(R) is well studied by Fabes and Rivière. Our goal is to extend the...

In this paper, using the concept of w-distances on a metric space, we first prove a generalized fixed point theorem for mappings without continuity in a complete metric space. Using this result, we obtain new and well-known fixed point theorems in a complete metric space.

We introduce a quantale-valued generalization of approach spaces in terms of quantale-valued gauges. The resulting category is shown to be topological and to possess an initially dense object. Moreover we show that the category of quantale-valued approach spaces defined recently in terms of quantale-valued closures is a coreflective subcategory of our category and, for certain choices of the qu...

In the context of possibly infinite computations yielding finite or infinite (binary) outputs, the space 2≤ω = 2∗∪2ω appears to be one of the most fundamental spaces in Computer Science. Though underconsidered, next to 2, this space can be viewed (§3.5.2) as the simplest compact space native to computer science. In this paper we study some of its properties involving topology and computability....