λ-Topologies on Function Spaces

This paper is devoted to the spaces C _λ(X) of all continuous real-valued functions on X endowed with arbitrary λ-topologies. This is a fairly complete survey of the results obtained by the author in the following domains of the theory of λ-topologies: cardinal functions; locally convex properties; weak and strong topologies; dual spaces; lattices of λ-topologies; completeness. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 2, pp. 3–56, 2003.     

On the topology induced by the adjoint of a semigroup of operators

The adjoint of aC ₀-semigroup on a Banach spaceX induces a locally convex σ(X,X ^ℴ)-topology inX, which is weaker than the weak topology ofX. In this paper we study the relation between these two topologies. Among other things a class of subsets ofX is given on which they coincide. As an application, an Eberlein-Shmulyan type theorem is proved for the σ(X,X ^ℴ)-topology and it is shown that the uniform limit of σ(X,X ^ℴ)-compact operators is σ(X,X ^ℴ)-compact. Finally our results are applied to the problem when the Favard class of a semigroup equals the domain of the infinitesimal generator.     

Classification of topologies on finite sets using graphs

With the use of directed graphs, we study topologies on finite sets. On this basis, we propose a new classification of these topologies. Some properties of T ₀-topologies on finite sets are proved. In particular, we prove the existence, in T ₀-topologies, of open sets containing any number of elements that does not exceed the cardinality of the set itself. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 7, pp. 992–996, July, 2008.     

Complete positivity,tensor products and C*-nuclearity for inverse limits of C*-algebras

The paper aims at developing a theory of nuclear (in the topological algebraic sense) pro-C^*-algebras (which are inverse limits of C^*-algebras) by investigating completely positive maps and tensor products. By using the structure of matrix algebras over a pro-C^*-algebra, it is shown that a unital continuous linear map between pro-C^*-algebrasA andB is completely positive iff by restriction, it defines a completely positive map between the C^*-algebrasb(A) andb(B) consisting of all bounded elements ofA andB. In the metrizable case,A andB are homeomorphically isomorphic iff they are matricially order isomorphic. The injective pro-C^*-topology α and the projective pro-C^*-topology v on A⊗B are shown to be minimal and maximal pro-C^*-topologies; and α coincides with the topology of biequicontinous convergence iff eitherA orB is abelian. A nuclear pro-C^*-algebraA is one that satisfies, for any pro-C^*-algebra (or a C^*-algebra)B, any of the equivalent requirements; (i) α =v onA ⊗B (ii)A is inverse limit of nuclear C^*-algebras (iii) there is only one admissible pro-C^*-topologyon A⊗B (iv) the bounded partb(A) ofA is a nuclear C⊗-algebra (v) any continuous complete state map A→B^* can be approximated in simple weak^* convergence by certain finite rank complete state maps. This is used to investigate permanence properties of nuclear pro-C^*-algebras pertaining to subalgebras, quotients and projective and inductive limits. A nuclearity criterion for multiplier algebras (in particular, the multiplier algebra of Pedersen ideal of a C^*-algebra) is developed and the connection of this C^*-algebraic nuclearity with Grothendieck’s linear topological nuclearity is examined. A σ-C^*-algebraA is a nuclear space iff it is an inverse limit of finite dimensional C^*-algebras; and if abelian, thenA is isomorphic to the algebra (pointwise operations) of all scalar sequences.     

Sur les espaces de Banach contenantl 1(τ)

Letτ be a cardinal with cf(τ)>ℵ₀. Then a Banach spaceE contains a subspace isomorphic tol ^l(τ) if and only if [0,1] ^r is a continuous image of the unit ballE′₁ ofE′, provided with the w*-topology. It follows that, for each cardinalκ, ifE′₁ contains a copy ofβκ, thenE has a quotient isomorphic tol ^∞(κ). In this situation we show thatE has even a quotientisometric tol ^∞(κ).      

Functional limit theorems for cumulative processes and stopping times

Summary Consider a cumulative regenerative process with increments between regeneration points being i.i.d. r.v.'s. Let the d.f. of those increments belong to the domain of attraction of a stable distribution with exponent less than two. A functional limit theorem in the Skorohod M ₁-topology is proved for this process. The M ₁-topology is more useful than the J ₁-topology in this case, because it allows the cumulative process to be continuous.The second part of the paper concerns a stopping time process, (t)--inf(s>0:w(s)>tg(s)), where w(t) is a process with positive drift for which a functional limit theorem holds and g(t)=t ^p L(t) with 0p<1 and L(t) varying slowly at infinity. Weak convergence for the process (t) is proved under certain conditions in the J ₁- and M ₁-topologies.     

Ruled surfaces of Weingarten type in Minkowski 3-space, II

In our previous paper of the same title, we did not study the ruled surfaces of Weingarten type M : x(s, t)=α(s)+t β (s) in Minkowski 3-space with vector fields β and β′ along the base curve β such that β is nowhere null but β′ is null everywhere. We here fulfill our project by investigating this remaining case.     

An oscillating multiplier on Herz type spaces

In this paper, the boundedness of an oscillating multiplier m γ,β for different β on the Herz type spaces is obtained. This operator was initially studied by Wainger and Fefferman-Stein. Our results extend one of the main results in a paper by Xiaochun Li and Shanzhen Lu for the non-weighted case, if β is close to 1 or α is suitably large. For β ≥ 1, the results with no weights on the Herz type spaces are also new.     

<Emphasis Type="Italic">β</Emphasis>-compactness in L-fuzzy topological spaces

In this paper, β-compactness is introduced in L-topological spaces, where L is a fuzzy lattice. And its topological properties are systematically studied. This β-compactness is defined for arbitrary L-subsets. The intersection of a β-compact L-fuzzy subset and a β-closed L-fuzzy subset is β-compact with a good extension. Also different characterizations are obtained and some of its properties are studied.     

Monotone perturbations of the laplacian inL 1(R N )

The semilinear perturbation of Poisson’s equation (E): −Δu+β(u)∋f, where β is a maximal monotone graph inR, has been investigated by Ph. Bénilan, H. Brézis and M. Crandall forf∈L ¹(R ^N ),N≧1, under the assumptions 0∈β(0) ifN≧3 and 0∈β(0) ∩ Int β(R) ifN=1,2. We discuss in this paper the solvability and well-posedness of (E) in terms of any maximal monotone graph β. In particular, if β takes only positive values andN≧3 we prove that no solution exists; ifN=2 we give necessary and sufficient conditions on β andf for (E) to be solvable in a natural sense.     

Simulated annealing with a potential function with discontinuous gradient on Rd

In this paper, we have proven that the simulated annealing processdX _t = −β(t) ∇U (X _t ) + √2dW _tt with a potential function on R^d, of which the gradient is discontinuous, converges in probability to a neighborhood of the global minima of the potential function.     

A family of second degree semi-classical forms of class <Emphasis Type="Italic">s</Emphasis>=1

Let u(β ₀) be the regular form fulfilling (u(β ₀))_2n+1=β ₀(u(β ₀))_2n ,n≥0 where β ₀ is an arbitrary complex number in such a way that for β ₀=0 one has the symmetric forms. Recently, the second degree symmetric semi-classical forms (when β ₀=0) of class s≤1 were determined. In this paper, we determine all the semi-classical forms u(β ₀) of class s=1, which are also of second degree when β ₀≠0.     

Weak and strong forms of β-irresolute functions

A subset A of a topological space X is said to be β-open [1] if A ⊂ Cl (Int (Cl (A))). A function f : X → Y is said to be β-irresolute [4] if for every β-open set V of Y, f ^-1(V) is β-open in X. In this paper we introduce weak and strong forms of β-irresolute functions and obtain several basic properties of such functions. This revised version was published online in August 2006 with corrections to the Cover Date.     

Semiclassical Resolvent Estimates for Schrödinger Operators with Coulomb Singularities

Consider the Schr?dinger operator with semiclassical parameter h, in the limit where h goes to zero. When the involved long-range potential is smooth, it is well known that the boundary values of the operator’s resolvent at a positive energy λ are bounded by O(h ⁻¹) if and only if the associated Hamilton flow is non-trapping at energy λ. In the present paper, we extend this result to the case where the potential may possess Coulomb singularities. Since the Hamilton flow then is not complete in general, our analysis requires the use of an appropriate regularization. Submitted: March 19, 2007. Accepted: March 3, 2008.     

Courants invariants par une action propre

The purpose of this paper is to characterise the invariant sections-distributions by a proper action. More precisely, we show that if G is a connected Lie group acting on a differentiable vector bundle E→V such that the induced action on V is proper, then the topological vector space of the G-invariant linear functionals (on the space of C ^∞ sections with compact support) equipped with the induced weak-topology (resp. the strong-topology), is isomorphic to the weak (resp. strong) topological dual of the space (of all G-invariant sections σ with compact quotient supp(σ)/G) equipped with a suitable topology; this coincides with the usual C ^∞-topology if the orbit space is compact, and with the Schwartz-topology if the group G is compact. Received: 8 June 1998 / Revised version: 22 September 1998     

Compactness for a class of hit-and-miss hyperspaces

In the study of some kind of generalized Vietoris-type topologies for the hyperspace of all nonempty closed subsets of a topological space (X, τ), namely the so called Δ-hit-and-miss-topologies with Δ⊇Cl(X) (or Δ-topologies), which was initiated by the second author in 1965, it is obvious, that the non-compactness of such a hyperspace often depends on the non-compactness even in the lower-semifinite topology (induced by the “hit-sets”), which is contained in all hypertopologies of this type. Otherwise, compactness for these topologies is easily obtained from the compactness of (X, τ) by well-known theorems, if the “miss-sets” are induced either by compact or closed subsets. To obtain a similar result for topologies with “miss-sets” generated by subsets with a property which generalizes both, closedness and compactness especially in the non-Hausdorff case, we use consequently a quite set-theoretical lemma, stated at the beginning.     

On property (β) in Banach lattices, Calderón-Lozanowskii and Orlicz-Lorentz spaces

The geometry of Calderón-Lozanowskii spaces, which are strongly connected with the interpolation theory, was essentially developing during the last few years (see [4, 9, 10, 12, 13, 17]). On the other hand many authors investigated property (β) in Banach spaces (see [7, 19, 20, 21, 25, 26]). The first aim of this paper is to study property (β) in Banach function lattices. Namely a criterion for property (β) in Banach function lattice is presented. In particular we get that in Banach function lattice property (β) implies uniform monotonicity. Moreover, property (β) in generalized Calderón-Lozanowskii function spaces is studied. Finally, it is shown that in Orlicz-Lorentz function spaces property (β) and uniform convexity coincide.     

Destroying cycles in digraphs

For a simple directed graph G with no directed triangles, let β(G) be the size of the smallest subset X ∈ E(G) such that G\X has no directed cycles, and let γ(G) denote the number of unordered pairs of nonadjacent vertices in G. Chudnovsky, Seymour, and Sullivan showed that β(G) ≤ γ(G), and conjectured that β(G) ≤ $\tfrac{{\gamma (G)}} {2}$\tfrac{{\gamma (G)}} {2} . In this paper we prove that β(G)<0.88γ(G).     

一类无限矩阵拓扑代数

Let λ and μ be sequence spaces and have both the signed-weak gliding hump property, (λ,μ) the algebra of the infinite matrix operators which transform λ into μ. In this paper, it is proved that if λ and μ are β-spaces and λ^β and ,μ^β have also the signed-weak gliding hump property, then for any polar topology τ, ((λ,μ),τ) is always sequentially complete locally convex topological algebra.     

Ergodicity of cylinder flows arising from irregularities of distribution

LetT be the mod 1 circle group, α∈T be irrational and 0<β<1. LetE be the closed subgroup ofR generated by β and 1. DefineX=T×E andT:X→X byT(x, t)=(x+α,t+1 _[0,β] (x)−β). Then we have the theorem:T is ergodic if and only if β is rational or 1, α and β are linearly independent over the rationals. This paper was prepared while I was very graciously hosted by the Centro de Investigacion y Estudios Avanzados, Mexico City.