On nearly abelian polynomial semigroups

Let be a polynomial semigroup containing an element with degree at least 2 with the semigroup operation being functional composition. We prove that is nearly abelian if and only if the smallest completely invariant closed subset of the Riemann sphere is not equal to the Riemann sphere. We also give a positive answer to Conjecture 7.1 in Hinkkanen and Martin's paper on the dynamics of semigroups of rational functions.

     

Stieltjes moment sequences and positive definite matrix sequences

For a certain constant (a little less than ), every function satisfying , , is a Stieltjes indeterminate Stieltjes moment sequence. For every indeterminate moment sequence there is a positive definite matrix sequence which is not of positive type and which satisfies , . For a certain constant (a little greater than ), for every function satisfying , , there is a convolution semigroup of measures on , with moments of all orders, such that , , and for every such convolution semigroup the measure is Stieltjes indeterminate for all .

     

On free semigroups of automaton transformations

It is established that the subset of freek-generated subsemigroups of the semigroup of all automaton transformations over a finite alphabet is a second category set (in the sense of the Baire category approach) in the set of allk-generated subsemigroups. A continuum series of pairs of automaton transformations each of which generates a free semigroup of rank two is indicated. A criterion is established for this semigroup to be a finite-automaton group. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 248–259, February, 1998. The author wishes to express his deep gratitude to Professor V. I. Sushchans'kii for permanent help and attention to the research. This research was partially supported by the ISSEP under grant No. GSU 051341.     

On finitely generated idempotent semigroups

We give two short proofs of the well-known fact that every finitely generated idempotent semigroup is finite.     

A note on the two-sided two-dimensional moment problem

It is shown that there is a positive semidefinite two-sided two-dimensional sequence f, which is not a moment sequence, such that log f(2m,2n) =O(m ² + n ²) as (m,n) →∞. This revised version was published online in June 2006 with corrections to the Cover Date.     

On sandwich sets and congruences on regular semigroups

Let S be a regular semigroup and E(S) be the set of its idempotents. We call the sets S(e, f)f and eS(e, f) one-sided sandwich sets and characterize them abstractly where e, f ∈ E(S). For a, a′ ∈ S such that a = aa′a, a′ = a′aa′, we call S(a) = S(a′a, aa′) the sandwich set of a. We characterize regular semigroups S in which all S(e; f) (or all S(a)) are right zero semigroups (respectively are trivial) in several ways including weak versions of compatibility of the natural order. For every a ∈ S, we also define E(a) as the set of all idempotets e such that, for any congruence ϱ on S, aϱa ² implies that aϱe. We study the restrictions on S in order that S(a) or be trivial. For , we define on S by a b if . We establish for which S are or congruences.     

On Perfect Subsemigroups of Q+

It has long been known that in order for a subsemigroup of Q ₊, properly containing {0}, to be perfect, it is necessary that it be non-C-finite. We define quasi-C-finite semigroups in such a way that it is necessary that the semigroup be non-quasi-C-finite. Every C-finite subsemigroup of Q ₊ is quasi-C- finite, but there is a quasi-C-finite subsemigroup of Q ₊ which is not C-finite. This revised version was published online in June 2006 with corrections to the Cover Date.     

Hankel operators and the Stieltjes moment problem

Let s be a non-vanishing Stieltjes moment sequence and let μ be a representing measure of it. We denote by μn the image measure in Cn of μ⊗σn under the map , where σn is the rotation invariant probability measure on the unit sphere. We show that the closure of holomorphic polynomials in L²(μn) is a reproducing kernel Hilbert space of analytic functions and describe various spectral properties of the corresponding Hankel operators with anti-holomorphic symbols. In particular, if n=1, we prove that there are nontrivial Hilbert-Schmidt Hankel operators with anti-holomorphic symbols if and only if s is exponentially bounded. In this case, the space of symbols of such operators is shown to be the classical Dirichlet space. We mention that the classical weighted Bergman spaces, the Hardy space and Fock type spaces fall in this setting.     

偏序半群的偏序扩张

引入了偏序半群(S,·,≤)上的半拟序σ及模σ半拟链的概念.通过模σ半拟链,将S的偏序≤扩张为≤*,讨论了(S,*,≤*)是偏序半群的充分条件,并获得了若干理想的结果.特别地,得到了SPO(S)到PO(S)的两个半格同态定理.最后,还给出了S的满足某些给定条件的有限子集在≤*下成链的充要条件.     

On Complexity of the Word Problem for Finitely Presented Commutative Semigroups

The word problem is tractable on a real-time Turing machine for every finitely presented commutative semigroup.     

Sebestyén moment problem: The multi-dimensional case

Given a family of vectors in a Hilbert space we characterize the existence of a family of commuting contractions on having regular dilation and such that

The theorem is a multi-dimensional analogue for some well-known operator moment problems due to Sebestyén in case or, recently, to Gavruta and Paunescu in case .

     

On the existence of chaotic and hypercyclic semigroups on Banach spaces

We prove that every separable infinite dimensional complex Banach space admits a hypercyclic uniformly continuous semigroup. We also prove that there exist Banach spaces admitting no chaotic strongly continuous semigroups.

     

The quadratic form in the Lévy-Khinchin formula on semigroups

In this paper we obtain the quadratic form in the Lévy-Khinchin formula on a commutative involutive semigroup, with a neutral element, as a sum of two simpler quadratic forms.

     

Further results in the theory of generalised inflations of semigroups

We determine the structure of semigroups that satisfy xyzw∈{xy,xw,zy,zw}. These semigroups are precisely those whose power semigroup is a generalised inflation of a band. The structure of generalised inflations of the following types of semigroups is determined: the direct product of a group and a band, a completely simple semigroup and a free semigroup F(X) on a set X. In the latter case the semigroup must be an inflation of F(X). We also prove that in any semigroup that equals its square, the power semigroup is a generalised inflation of a band if and only if it is an inflation of a band.     

On perturbations preserving the immediate norm continuity of semigroups

We show that the Desch-Schappacher perturbation and the Miyadera-Voigt perturbation of an immediately norm continuous semigroup are immediately norm continuous. We also show that a perturbation theorem of C. Batty, C. Kaiser and L. Weis based on a generation theorem of A.M. Gomilko, D.-X. Feng and D.-H. Shi also preserves the immediate norm continuity of semigroups. The novelty of these results is that, contrary to the numerous related results, we obtain the immediate norm continuity of the perturbed semigroup without additional assumptions.     

Linear continuous operators for the Stieltjes moment problem in Gelfand–Shilov spaces

We obtain linear continuous operators providing a solution to the Stieltjes moment problem in the framework of Gelfand–Shilov spaces of rapidly decreasing smooth functions. The construction rests on an interpolation procedure due to R. Estrada for general rapidly decreasing smooth functions, and adapted by S.-Y. Chung, D. Kim and Y. Yeom to the case of Gelfand–Shilov spaces. It requires a linear continuous version of the so-called Borel–Ritt–Gevrey theorem in asymptotic theory.     

On the construction of the conjugate hull of Brandt semigroups

In this paper the characterizations of conjugate hulls ψ (S). ϕ(S),T(S) and θ(S) on a Brandt semigroupS are given. By using these results we can pro thatT(S) is self-conjugate in ψ(S) for a Brandt semigroupS. This work is supported by National Natural Science Foundation of China