Class Of Semigroups Research Articles

Gevrey Class Semigroups Arising from Elastic Systems with Gentle Dissipation: The Case 0 < α < ½

Let A (the elastic operator) be a positive, self-adjoint operator with domain D(A) in the Hilbert space X, and let B (the dissipation operator) be another positive, self-adjoint operator satisfying ρ 1 A α 1/2α, hence differentiable on E for all t>0

Estimates for the fractal dimension and the number of determining modes for invariant sets of dynamical systems

Majorants of the fractal dimension and of the number of determining modes for unbounded sets, invariant with respect to operators of semigroups of classes 1 and 2, are obtained. They are computed for the Navier-Stokes equations (two- and three-dimensional) under the first boundary condition and under periodicity conditions in the spaces\(H_0 \subset \vec L_2 (\Omega )\) and\(H^1 \subset \vec W_2 {}^1(\Omega )\).

Semigroups whose idempotents form a subsemigroup

We prove that if the “type-II-construct” subsemigroup of a finite semigroup S is regular, then the “type-II” subsemigroup of S is computable (actually in this case, type-II and type-II-construct are equal). This, together with certain older results about pseudo-varieties of finite semigroups, leads to further results:(1) We get a new proof of Ash's theorem: If the idempotents in a finite semigroup S commute, then S divides a finite inverse semigroup. Equivalently: The pseudo-variety generated by the finite inverse semigroups consists of those finite semigroups whose idempotents commute.(2) We prove: If the idempotents of a finite semigroup S form a subsemigroup then S divides a finite orthodox semigroup. Equivalently: The pseudo-variety generated by the finite orthodox semigroups consists of those finite semigroups whose idempotents form a subsemigroup.(3) We prove: The union of all the subgroups of a semigroup S forms a subsemigroup if and only if 5 belongs to the pseudo-variety u * G if and only if Sn belongs to u. Here u denotes the pseudo-variety of finite semigroups which are unions of groups.For these three classes of semigroups, type-II is equal to type-II construct.

Rees matrix semigroups

In this paper we provide a new, abstract characterisation of classical Rees matrix semigroups over monoids with zero. The corresponding abstract class of semigroups is obtained by abstracting a number of algebraic properties from completely 0-simple semigroups: in particular, the relationship between arbitrary elements and idempotents.

Gevrey class semigroups arising from elastic systems with gentle dissipation: the case 0<𝛼<\frac12

LetAA(the elastic operator) be a positive, self-adjoint operator with domainD(A)\mathcal {D}(A)in the Hilbert spaceXX, and letBB(the dissipation operator) be another positive, self-adjoint operator satisfyingρ1Aα≤B≤ρ2Aα{\rho _1}{A^\alpha } \leq B \leq {\rho _2}{A^\alpha }for some constants0>ρ1>ρ2>∞0 > {\rho _1} > {\rho _2} > \inftyand0>α≤10 > \alpha \leq 1. Consider the operator\[AB=|0amp;I−Aamp;−B|{\mathcal {A}_B} = \left | {\begin {array}{*{20}{c}} 0 & I \\ { - A} & { - B} \\ \end {array} } \right |\](corresponding to the elastic modelx¨+Bx˙+Ax=0\ddot x + B\dot x + Ax = 0written as a first order system), which (once closed) is plainly the generator of a strongly continuous semigroup of contractions on the spaceE=D(A1/2)×XE = \mathcal {D}({A^{1/2}}) \times X. In [C-T.l] [C-T.3] we showed that, for12≤α≤1\tfrac {1}{2} \leq \alpha \leq 1, such a semigroup is analytic (holomorphic) onEEon a triangular sector ofC{\mathbf {C}}containing the positive real axis and, moreover, that the property of analyticity is false for0>α>120 > \alpha > \tfrac {1}{2}, say forB=AαB = {A^\alpha }. We now complete the description ofAB{\mathcal {A}_B}in the range0>α>120 > \alpha > \tfrac {1}{2}by showing that such semigroup is in fact of Gevrey classδ>1/2α\delta > 1/2\alpha, hence differentiable onEEfor allt>0t > 0.

Exponentially bounded C-semigroups and generation of semigroups

This paper concerns exponentially bounded C-semigroups and semigroups of operators in a Banach space X. Recently Davies and Pang [l] introduced the notion of an exponentially bounded C-semigroup and characterized the generator of an exponentially bounded C-semigroup. On the other hand, Miyadera [7] discussed the complete infinitesimal generator of an exponentially bounded C-semigroup, and Tanaka [14] has characterized the complete infinitesimal generator of an exponentially bounded C-semigroup. There are many important works on semigroups of linear operators. (For example, see [4, 5, 12, 161.) Extending the notion of (C,)-semigroups, Oharu [9] introduced the semigroups of classes (Cc,,), k = 0, 1,2, . . . . and Da Prato [2] and Sobolevskii [13] introduced the semigroups of growth order c( 2 0. Generation theorems for semigroups of these classes have been studied in [2, 9, 10, 131. Our first purpose is to clarify the relationship between semigroups and exponentially bounded C-semigroups. It is shown that if {T(t); t 2 0} is a semigroup and C is an injective bounded linear operator with dense range satisfying r(t)C= CT(t) for t > 0 and R(C) c C= (xEX; lim r-o+ T(r)x = x}, then {CT(t); t 2 0) is an exponentially bounded C-semigroup whose complete infinitesimal generator is equal to that of {T(t); t 2 0). In particular, if { r(t); t k 0} is a semigroup of class (C,,,) or of growth order ~1, then {CT(t); t 2 O> becomes an exponentially bounded C-semigroup with a suitable C and its complete infinitesimal generator coincides with that of {T(t); t 2 O}. Our second and main purpose is to study the generation of semigroups via the theory of exponentially bounded C-semigroups. We shall characterize the complete infinitesimal generator of a semigroup {r(t); t 2 0}

A note on the convergence of linear semi-groups of class (1,A)

A note on the convergence of linear semi-groups of class (1,A)

Stabilization of Kurosh chains in the class of semigroups with zero

Stabilization of Kurosh chains in the class of semigroups with zero

Perturbation of the generating operator of an analytic semigroup

for sufficiently small positive b. For differential operators this condition means that the perturbing operator has smaller order than the operator A or their orders coincide but the coefficients of the operator Q are small in comparison with the coefficients of the operator A. It turns out that the operator A + Q generates an analytic semigroup also in the case when condition (i) is not satisfied, if one assumes that the operator Q is the generating operator of a strongly continuous semigroup of class c o and the commutator A(A, Q) = AQ QA satisfies some subordination condition. The present note is devoted to the proof of this statement. We make the following assumptions regarding the operators A and Q. I. D(A) = D(Q) = D~, ~i = E. II. The operator -A is strongly positive in the spaces E and D~. III. The operator Q generates in the spaces E and Dz a strongly continuous semigroup of class c o . IV. The operators QA -z and Q2A-2 are bounded in the space E. V. There exists p > 0 such that the operator A(A,Q)(--A) -(2-p) admits a closure to a bounded operator in the spaces E and DI. We mention that condition V is satisfied, for example, for an elliptic operator A of order 2m with smooth coefficients and a differential operator Q of order k (k< 2m), acting in the space Lp(R n) if 0<~<i/2m. ~ We give preliminary arguments which allow us to obtain an integral equation, whose solution is the semigroup generated by the operator A + Q. We make use of a statement established in [2]. LEMMA. Assume that the operators B, C, and B + C generate strongly continuous semigroups. Then we have the equality

A Cancellation Theorem for Modules Over the Group C*-Algebras of Certain Nilpotent Lie Groups

In recent years, there has been a rapid growth of the K-theory of C*-algebras. From a certain point of view, C*-algebras can be treated as “non-commutative topological spaces”, while finitely generated projective modules over them can be thought of as “non-commutative vector bundles”. The K-theory of C*-algebras [30] then generalizes the classical K-theory of topological spaces [1]. In particular, the K0-group of a unital C*-algebra A is the group “generated” by (or more precisely, the Grothendieck group of) the commutative semigroup of stable isomorphism classes of finitely generated projective modules over A with direct summation as the binary operation. The semigroup gives an order structure on K0(A) and is usually called the positive cone in K0(A).Around 1980, the work of Pimsner and Voiculescu [18] and of A. Connes [4] provided effective ways to compute the K-groups of C*-algebras. Then the classification of finitely generated projective modules over certain unital C*-algebras up to stable isomorphism could be done by computing their K0-groups as ordered groups. Later on, inspired by A. Connes's development of non-commutative differential geometry on finitely generated projective modules [2], the deeper question of classifying such modules up to isomorphism and hence the so-called cancellation question were raised (cf. [21] ).

Bistable orderings on a certain class of semigroups

Bistable orderings on a certain class of semigroups

Evolution operators related to semi-groups of class (A)

Evolution operators related to semi-groups of class (A)

Dense subsets of some common classes of semigroup

Dense subsets of some common classes of semigroup

Uniform ergodic theorems for locally integrable semigroups and pseudoresolvents

We study uniform ergodicity (at ∞ \infty ) of a locally integrable operator semigroup T ( ⋅ ) T( \cdot ) of type w 0 {w_0} under a suitable condition which is weaker than the usual one w 0 ⩽ 0 {w_0} \leqslant 0 . We also give a precise characterization of the uniform Cesàro-ergodicity for semigroups of class ( 0 , A ) (0,A) . To prove the part of Abel-ergodicity we first prove a general uniform ergodic theorem for pseudo-resolvents.

Abelian semigroups whose Stone-Čech compactifications have left ideal decompositions

Stone-Čech compactifications of semigroups have aroused a good deal of interest recently. Several authors, for example Milnes [6], Marcri [5] and Baker and Butcher [1], have concentrated on problems of the existence of continuous extensions to βSof the operation in a topological semigroupS.For a discrete semigroup a separately continuous extension always exists, and others such as Pym and Vasudeva[8] have studied the compactifications of particular classes of semigroups. Further interest has centred on the algebraic structure of these compactifications; see for example Hindman[3].

On approximation by operator semigroups of a general type

For operator semigroups of class (C0) on a Banach space X it is well known that the saturation class can be characterized as the relative completion with respect to X of the domain of the infinitesimal generator. This remains true for strongly measurable semigroups {T(t),t>0} having a closed infinitesimal operator A0, but it becomes false if A0 is non-closed. We prove that a characterization is given by ∥A0T(t)f ∥=0(1), t→0 + for a fairly general class of semigroups, including certain particular semigroups which belong to Oharu's class (C(1)), or are of growth order less than one.

Globally determined classes of semigroups

Globally determined classes of semigroups

Nondegenerate symmetric bilinear forms on finite abelian 2-groups

Let B 2 {\mathcal {B}_2} be the semigroup of isomorphism classes of finite abelian 2 2 -groups with a nondegenerate symmetric bilinear form having values in Q / Z Q/{\mathbf {Z}} . Generators for B 2 {\mathcal {B}_2} were given by C. T. C. Wall and the known relations among these generators were proved to be complete by A. Kawauchi and S. Kojima. In this article we describe a normal form for such bilinear forms, expressed in terms of Wall’s generators, and as a by-product we obtain a simpler proof of the completeness of the known relations.

Epis are onto for finite regular semigroups

After preliminary results and definitions in Section 1, we show in Section 2 that any finite regular semigroup is saturated, in the sense of Howie and Isbell [8] (that is, the dominion of a finite regular semigroup U in a strictly containing semigroup S is never S). This is equivalent of course to showing that in the category of semigroups any epi from a finite regular semigroup is in fact onto. Note for inverse semigroups the stronger result, that any inverse semigroup is absolutely closed [11, Theorem VII. 2.14] or [8, Theorem 2.3]. Further, any inverse semigroup is in fact an amalgamation base in the class of semigroups [10], in the sense of [5]. These stronger results are known to be false for finite regular semigroups [8, Theorem 2.9] and [5, Theorem 25]. Whether or not every regular semigroup is saturated is an open problem.

Holomorphicity of a Class of Semigroups of Measures Operating on L p (G/H)

Holomorphicity of a Class of Semigroups of Measures Operating on L p (G/H)