On subsemigroups of βℕ and absolute coretracts

A semigroup S is called an absolute coretract if for any continuous homomorphism f from a compact Hausdorff right topological semigroup T onto a compact Hausdorff right topological semigroup containing S algebraically there exists a homomorphism g \colon S→ T such that f\circ g=id _S . The semigroup β\ben contains isomorphic copies of any countable absolute coretract. In this article we define a class C of semigroups of idempotents each of which is a decreasing chain of rectangular semigroups. It is proved that every semigroup from C is an absolute coretract and every finite semigroup of idempotents, which is an absolute coretract, belongs to C . July 25, 2000     

Automorphisms of linear semigroups over division rings

Given an-dimensional right vector spaceV over a division ring we denote byS the semigroup of the endomorphisms ofV and designate this semigroup as alinear semigroup. First we prove that every automorphism ofS can be written asT→fTf ⁻¹, wheref∶V→V is a semilinear homeomorphism. Furthermore, we show that every isomorphism between maximal compact subsemigroups ofS is also of this type.     

Continuation of a linear operator to an involution operator

A bounded linear operatorA:X→X in a linear topological spaceX is called ap-involution operator,p≥2, ifA ^p=I, whereI is the identity operator. In this paper, we describe linearp-involution operators in a linear topological space over the field ℂ and prove that linear operators can be continued to involution operators. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 671–676, May, 1997. Translated by M. A. Shishkova     

Weak convergence of trajectories of nonexpansive semigroups in Hilbert space

LetS: I×X→X be a nonexpansive semigroup on a weakly closed (not necessarily convex) subsetX of a real Hilbert spaceE. In this note we present a theorem on the weak convergence of a trajectory {S(t,x)} _t∈I together with a very simple and elementary proof, which extends and unifies several recent results due to Baillon, Bruck, Pazy and Reich.     

Some properties and applications of weakly equicompact sets

Let X and Y be Banach spaces. A set (the space of all weakly compact operators from X into Y) is weakly equicompact if, for every bounded sequence (x _n) in X, there exists a subsequence (x _k(n)) so that (Tx_k(n)) is uniformly weakly convergent for T ∈ M. In this paper, the notion of weakly equicompact set is used to obtain characterizations of spaces X such that X ↩̸ ℓ₁, of spaces X such that B _X* is weak* sequentially compact and also to obtain several results concerning to the weak operator and the strong operator topologies. As another application of weak equicompactness, we conclude a characterization of relatively compact sets in when this space is endowed with the topology of uniform convergence on the class of all weakly null sequences. Finally, we show that similar arguments can be applied to the study of uniformly completely continuous sets. Received: 5 July 2006     

Polyhedral banach spaces and extensions of compact operators

LetX be a polyhedral Banach space whose dual is anL ₁(μ) space for some measureμ. Then for each Banach spacesY ⊆Z and each compact operatorT: Y →X there exists a norm preserving compact extension Z →X.     

Semigroups of quasi-compact operators

LetT ₁ andT ₂ be two quasi-compact operators on a complex Banach spaceX, whereX is the Banach space of all complex valued and continuous functions on a compact Hausdorff spaceY. LetS(I, T ₁,T ₂) denote the semigroup generated byT ₁,T ₂ and the identity operatorI. We show that under certain conditions the kernel of the semigroup is a finite group. I am grateful to Professor C. T. Taam, my advisor at The George Washington University. His support and his valuable contributions made this research possible. I am also very thankful to the referee for many important suggestions which improved the results and quality of the paper. In particular, the proof in Theorem 4 which shows the kernel of is a finite group, has been suggested by the referee.     

<Emphasis Type="Italic">k</Emphasis>*-Metrizable spaces and their applications

In this paper, we introduce and study a new class of generalized metric spaces, which we call k*-metrizable spaces, and suggest various applications of such spaces in topological algebra, functional analysis, and measure theory. By definition, a Hausdorff topological space X is k*-metrizable if X is the image of a metrizable space M under a continuous map f: M → X which has a section s: X → M preserving precompact sets in the sense that the image s(K) of any compact set K ⊂ X has compact closure in X. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 48, General Topology, 2007.     

The product of the idempotents and an $\mathcal{H}$-class of the finite full transformation semigroup

We investigate the set products S=EH, where E is the set of idempotents of a finite full transformation semigroup T _X and H is an arbitrary H\mathcal{H}-class of T _X . We show that S is a semigroup and is a union of H\mathcal{H}-classes of T _X . We determine the nature of this union through use of Hall’s Marriage Lemma. We describe Green’s relations and thereby show that S has regular elements of all possible ranks and that \operatornameReg(S)\operatorname{Reg}(S) forms a right ideal of S.     

A non-reflexive Grothendieck space that does not containl ∞

A compact spaceS is constructed such that, in the dual Banach spaceC(S)^*, every weak^* convergent sequence is weakly convergent, whileC(S) does not have a subspace isomorphic tol _∞. The construction introduces a weak version of completeness for Boolean algebras, here called the Subsequential Completeness Property. A related construction leads to a counterexample to a conjecture about holomorphic functions on Banach spaces. A compact spaceT is constructed such thatC(T) does not containl _∞ but does have a “bounding” subset that is not relatively compact. The first of the examples was presented at the International Conference on Banach spaces, Kent, Ohio, 1979.     

The uniform approximation property in Orlicz spaces

It is proved that for every reflexive Orlicz spaceX there is a functionn(k,ε) so that wheneverE is ak-dimensional subspace ofX there exists an operatorT: X→X such thatT _1E=identity, ‖T‖≦1+ε and dimTX≦n(k,ε). Some general facts concerning the uniform approximation property are also presented. Research of the first named author was partially supported by NSF Grant MPS 74-07509-A01.     

A structure theorem for boundary-transitive graphs with infinitely many ends

We prove a structure theorem for locally finite connected graphsX with infinitely many ends admitting a non-compact group of automorphisms which is transitive in its action on the space of ends, Ω _X . For such a graphX, there is a uniquely determined biregular treeT (with both valencies finite), a continuous representationφ : Aut(X) → Aut(T) with compact kernel, an equivariant homeomorphism λ : Ω _X → Ω _T , and an equivariant map τ : Vert(X) → Vert(T) with finite fibers. Boundary-transitive trees are described, and some methods of constructing boundary-transitive graphs are discussed, as well as some examples.     

Extremal properties of contraction semigroups onc 0

For any complex Banach spaceX, letJ denote the duality mapping ofX. For any unit vectorx inX and any (C ₀) contraction semigroup (T _t ) _t>0 onX, Baillon and Guerre-Delabriere proved that ifX is a smooth reflexive Banach space and if there isx ^*∈J(x) such that ÷〈(T(t)x, J(x)〈÷→1 ast→∞, then there is a unit vectory∈X which is an eigenvector of the generatorA of (T _t ) _t>0 associated with a purely imaginary eigenvalue. They asked whether this result is still true ifX is replaced byc ₀. In this article, we show the answer is negative Partial results of this paper were obtained when the author attended the International Conference of Convexity at the University of Marne-La-Vallée. He would like to express his gratitude for the kind hospitality offered to him. He would also like to thank Profs. Goldstein and Jamison for their valuable suggestions.     

The space of maps from a locally compact space to a Banach space

The space of continuous maps from a topological spaceX to topological spaceY is denoted byC(X,Y) with the compact-open topology. In this paper we prove thatC(X,Y) is an absolute retract ifX is a locally compact separable metric space andY a convex set in a Banach space. From the above fact we know thatC(X,Y) is homomorphic to Hilbert spacel ₂ ifX is a locally compact separable metric space andY a separable Banach space; in particular,C(R ⁿ,R^m) is homomorphic to Hilbert spacel ₂. This research is supported by the Science Foundation of Shanxi Province's Scientific Committee     

On perturbations of differentiable semigroups

LetX be a Banach space and letA be the infinitesimal generator of a differentiable semigroup {T(t) |t ≥ 0}, i.e. aC ₀-semigroup such thatt ↦T(t)x is differentiable on (0, ∞) for everyx εX. LetB be a bounded linear operator onX and let {S(t) |t ≥ 0} be the semigroup generated byA +B. Renardy recently gave an example which shows that {S(t) |t ≥ 0} need not be differentiable. In this paper we give a condition on the growth of ‖T′(t)‖ ast ↓ 0 which is sufficient to ensure that {S(t) |t ≥ 0} is differentiable. Moreover, we use Renardy’s example to study the optimality of our growth condition. Our results can be summarized roughly as follows:

Extending an operator from a Hilbert space to a larger Hilbert space,so as to reduce its spectrum

In our earlier paper [1] we showed that given any elementx of a commutative unital Banach algebraA, there is an extensionA′ ofA such that the spectrum ofx inA′ is precisely the essential spectrum ofx inA. In [2], we showed further that ifT is a continuous linear operator on a Banach spaceX, then there is an extensionY ofX such thatT extends continuously to an operatorT ⁻ onY, and the spectrum ofT ⁻ is precisely the approximate point spectrum ofT. In this paper we take the second of these results, and show further that ifX is a Hilbert space then we can ensure thatY is also a Hilbert space; so any operatorT on a Hilbert spaceX is the restriction to one copy ofX of an operatorT ⁻ onX ⊕X, whose spectrum is precisely the approximate point spectrum ofT. This result is “best possible” in the sense that if isany extension to a larger Banach space of an operatorT, it is a standard exercise that the approximate point spectrum ofT is contained in the spectrum of .     

A theorem of Riesz type with Pettis integrals in topological vector spaces

Let X be a locally convex Hausdorff space and let C₀(S,X) be the space of all continuous functions f:S→X, with compact support on the locally compact space S. In this paper we prove a Riesz representation theorem for a class of bounded operators T:C₀(S,X)→X, where the representing integrals are X-valued Pettis integrals with respect to bounded signed measures on S. Under the additional assumption that X is a locally convex space, having the convex compactness property, or either, X is a locally convex space whose dual X^′ is a barrelled space for an appropriate topology, we obtain a complete identification between all X-valued Pettis integrals on S and the bounded operators T:C₀(S,X)→X they represent. Finally we give two illustrations of the representation theorem proved, in the particular case when X is the topological dual of a locally convex space.     

Multivalued perturbations ofm-accretive differential inclusions

Given anm-accretive operatorA in a Banach spaceX and an upper semicontinuous multivalued mapF: [0,a]×X→2 ^X , we consider the initial value problemu′∈−Au+F(t,u) on [0,a],u(0)=x ₀. We concentrate on the case when the semigroup generated by—A is only equicontinuous and obtain existence of integral solutions if, in particular,X* is uniformly convex andF satisfies β(F(t,B))≤k(t)β(B) for all boundedB⊂X wherek∈L ¹([0,a]) and β denotes the Hausdorff-measure of noncompactness. Moreover, we show that the set of all solutions is a compactR _δ-set in this situation. In general, the extra condition onX* is essential as we show by an example in whichX is not uniformly smooth and the set of all solutions is not compact, but it can be omited ifA is single-valued and continuous or—A generates aC _o-semigroup of bounded linear operators. In the simpler case when—A generates a compact semigroup, we give a short proof of existence of solutions, again ifX* is uniformly (or strictly) convex. In this situation we also provide a counter-example in ℝ⁴ in which no integral solution exists. The author gratefully acknowledges financial support by DAAD within the scope of the French-German project PROCOPE.     

Topological dynamics of transformations induced on the space of probability measures

LetT be a continuous transformation of a compact metric spaceX. T induces in a natural way a transformationT _M on the spaceM (X) of probability measures onX, and a transformationT _K on the spaceK (X) of closed subsets ofX. This note investigates which of the topological properties ofT∶X→X (like distality, transitivity, mixing property etc. ...) are “inherited” byT _M∶M (X)→M (X) andT _K∶K (X)→K (X).     

The continuity of the inversion and the structure of maximal subgroups in countably compact topological semigroups

We search for conditions on a countably compact (pseudocompact) topological semigroup under which: (i) each maximal subgroup H(e) in S is a (closed) topological subgroup in S; (ii) the Clifford part H(S) (i.e. the union of all maximal subgroups) of the semigroup S is a closed subset in S; (iii) the inversion inv: H(S) → H(S) is continuous; and (iv) the projection π: H(S) → E(S), π: x ↦ xx ⁻¹, onto the subset of idempotents E(S) of S, is continuous.