Perturbations of Bi-continuous Semigroups with Applications to Transition Semigroups on C<Subscript>b</Subscript>(H)

We prove an unbounded perturbation theorem for bi-continuous semigroups on the space of bounded, continuous functions on the Hilbert space H. This is applied to the Ornstein-Uhlenbeck semigroup, thus providing a purely functional analytic approach to the existence of transition semigroups on C_b(H) with bounded non-linear drift.     

Semigroup of matrices acting on the max-plus projective space

We investigate the action of semigroups of d×d matrices with entries in the max-plus semifield on the max-plus projective space. Recall that semigroups generated by one element with projectively bounded image are projectively finite and thus contain idempotent elements.In terms of orbits, our main result states that the image of a minimal orbit by an idempotent element of the semigroup with minimal rank has at most d! elements. Moreover, each idempotent element with minimal rank maps at least one orbit onto a singleton.This allows us to deduce the central limit theorem for stochastic recurrent sequences driven by independent random matrices that take countably many values, as soon as the semigroup generated by the values contains an element with projectively bounded image.     

Robustness of strongly and polynomially stable semigroups

In this paper we study the robustness properties of strong and polynomial stability of semigroups of operators. We show that polynomial stability of a semigroup is robust with respect to a large and easily identifiable class of perturbations to its infinitesimal generator. The presented results apply to general polynomially stable semigroups and bounded perturbations. The conditions on the perturbations generalize well-known criteria for the preservation of exponential stability of semigroups. We also show that the general results can be improved if the perturbation is of finite rank or if the semigroup is generated by a Riesz-spectral operator. The theory is applied to deriving concrete conditions for the preservation of stability of a strongly stabilized one-dimensional wave equation.     

Convergence of operator semigroups associated with generalised elliptic forms

In a recent article, Arendt and ter Elst have shown that every sectorial form is in a natural way associated with the generator of an analytic strongly continuous semigroup, even if the form fails to be closable. As an intermediate step, they have introduced so-called j?elliptic forms, which generalise the concept of elliptic forms in the sense of Lions. We push their analysis forward in that we discuss some perturbation and convergence results for semigroups associated with j?elliptic forms. In particular, we study convergence with respect to the trace norm or other Schatten norms. We apply our results to Laplace operators and Dirichlet-to-Neumann-type operators.     

The Word Problem for Relatively Free Semigroups

An example of a series of varieties of semigroups X_p with the finite basis property is constructed for which the word problem in the relatively free semigroup F_n X_p of rank n in the variety X_p is decidable if and only if n < p.     

Invariant Subspaces for Semigroups of Algebraic Operators

T. Laffey showed (Linear and Multilinear Algebra6(1978), 269–305) that a semigroup of matrices is triangularizable if the ranks of all the commutators of elements of the semigroup are at most 1. Our main theorem is an extension of Hthis result to semigroups of algebraic operators on a Banach space. We also obtain a related theorem for a pair {A, B} of arbitrary bounded operators satisfying rank (AB−BA)=1 and several related conditions. In addition, it is shown that a semigroup of algebraically unipotent operators of bounded degree is triangularizable.     

Semigroup closures of finite rank symmetric inverse semigroups

We introduce the notion of semigroup with a tight ideal series and investigate their closures in semitopological semigroups, particularly inverse semigroups with continuous inversion. As a corollary we show that the symmetric inverse semigroup of finite transformations I _λ ⁿ of the rank ≤ n is algebraically closed in the class of (semi)topological inverse semigroups with continuous inversion. We also derive related results about the nonexistence of (partial) compactifications of classes of semigroups that we consider.     

Essential growth rate for bounded linear perturbation of non-densely defined Cauchy problems

This paper is devoted to the study of the essential growth rate of some class of semigroup generated by bounded perturbation of some non-densely defined problem. We extend some previous results due to Thieme [H.R. Thieme, Quasi-compact semigroups via bounded perturbation, in: Advances in Mathematical Population Dynamics—Molecules, Cells and Man, Houston, TX, 1995, in: Ser. Math. Biol. Med., vol. 6, World Sci. Publishing, River Edge, NJ, 1997, pp. 691-711] to a class of non-densely defined Cauchy problems in Lp. In particular in the context the integrated semigroup is not operator norm locally Lipschitz continuous. We overcome the lack of Lipschitz continuity of the integrated semigroup by deriving some weaker properties that are sufficient to give information on the essential growth rate.     

On the Relation between Stability of Continuous- and Discrete-Time Evolution Equations via the Cayley Transform

In this paper we investigate and compare the properties of the semigroup generated by A, and the sequence where A_d = (I + A) (I − A)⁻¹. We show that if A and A⁻¹ generate a uniformly bounded, strongly continuous semigroup on a Hilbert space, then A_d is power bounded. For analytic semigroups we can prove stronger results. If A is the infinitesimal generator of an analytic semigroup, then power boundedness of A_d is equivalent to the uniform boundedness of the semigroup generated by A.     

Finite semigroups with slowly growing pn-sequences

The p _n -sequence of a semigroup S is said to be polynomially bounded, if there exist a positive constant c and a positive integer r such that the inequality p _n (S) ≤cn ^r holds for all n≥ 1. In this paper, we fully describe all finite semigroups having polynomially bounded p _n -sequences. First we give a characterization in terms of identities satisfied by these semigroups. In the sequel, this result will allow an insight into the structure of such semigroups. We are going to deal with certain ideals and the construction of ideal extension of semigroups. In addition, we supply an effective procedure for deciding whether a finite semigroup has polynomially bounded p _n -sequence and give some examples. Received March 5, 1999; accepted in final form November 1, 1999.     

Adjoint bi-continuous semigroups and semigroups on the space of measures

For a given bi-continuous semigroup (T(t)) _t⩾0 on a Banach space X we define its adjoint on an appropriate closed subspace X° of the norm dual X′. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology σ(X°,X). We give the following application: For Ω a Polish space we consider operator semigroups on the space C_b(Ω) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(Ω) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(Ω) are precisely those that are adjoints of bi-continuous semigroups on C_b(Ω). We also prove that the class of bi-continuous semigroups on C_b(ω) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if is not a Polish space this is not the case.     

Ergodic properties of operator semigroups in general weak topologies

The aim of this paper is to study ergodic properties (i.e., properties about the limit of Cesàro averages) of a semigroup of bounded linear operators on a Banach space X, which is assumed to be continuous and locally integrable in the sense of a certain general weak topology of X. Then the results are applied to particular examples, such as locally strongly integrable semigroups, their dual semigroups, and the tensor product semigroup of two (C₀)-semigroups.     

Commutative cancellative semigroups of finite rank

The rank of a commutative cancellative semigroup S is the cardinality of a maximal independent subset of S. Commutative cancellative semigroups of finite rank are subarchimedean and thus admit a Tamura-like representation. We characterize these semigroups in several ways and provide structure theorems in terms of a construction akin to the one devised by T. Tamura for N-semigroups.     

A perturbation theorem for the essential spectral radius of strongly continuous semigroups

We generalize the Jörgens-Vidav semigroup perturbation theorem: If (U (t)), (V (t)) are s. c. semigroups, the generator of (V (t)) a bounded perturbation of the generator of (U (t)), and some remainder in the iteration series ofV (t) is strictly power compact, then the spectrum ofV (t) outside the spectral disc ofU (t) consists of eigenvalues of finite algebraic multiplicity. As a prerequisite, we show the invariance of components of the essential resolvent set of an operator under relatively power compact pertubations.     

Dual semigroups and second order linear elliptic boundary value problems

It is shown that general second order elliptic boundary value problems on bounded domains generate analytic semigroups onL ₁. The proof is based on Phillips’ theory of dual semigroups. Several sharp estimates for the corresponding semigroups inL _p, 1≦p<∞, are given.     

Duality relation for the Hilbert series of almost symmetric numerical semigroups

We derive the duality relation for the Hilbert series H (d ^m ; z) of the almost symmetric numerical semigroup S (d ^m ) combining it with its dual H (d ^m ; z ⁻¹). We establish the bijection between the multiset of degrees of the syzygy terms and the multiset of the gaps F _j , generators d _i and their linear combinations. We present the relations for the sums of the Betti numbers of even and odd indices separately. We apply the duality relation to the simple case of the almost symmetric semigroups of maximal embedding dimension, and give the necessary and sufficient conditions for the minimal set d ^m to generate such semigroups.     

The L ∞-Stokes semigroup in exterior domains

The Stokes semigroup on a bounded domain is an analytic semigroup on spaces of bounded functions as was recently shown by the authors based on an a priori L ^∞-estimate for solutions to the linear Stokes equations. In this paper, we extend our approach to exterior domains and prove that the Stokes semigroup is uniquely extendable to an analytic semigroup on spaces of bounded functions.     

Free <Emphasis Type="Italic">n</Emphasis>-Tuple Semigroups

The present paper is devoted to the study of n-tuple semigroups. A free n-tuple semigroup of arbitrary rank is constructed and, as a consequence, singly generated free n-tuple semigroups are characterized. Moreover, examples of n-tuple semigroups are presented, the independence of the n-tuple semigroup axioms is proved, and it is shown that the natural semigroups of the constructed free n-tuple semigroup are isomorphic and the automorphism group of this n-tuple semigroup is isomorphic to a symmetric group.     

Semigroup varieties with a small number of term operations

We give an equational characterization of (varieties of) semigroups having a p_n-sequence bounded above by a polynomial function of n. This is achieved by studying the syntactical connections between certain semigroup identities and their equational consequences.     

Convoluted C-cosine functions and semigroups. Relations with ultradistribution and hyperfunction sines

Convoluted C-cosine functions and semigroups in a Banach space setting extending the classes of fractionally integrated C-cosine functions and semigroups are systematically analyzed. Structural properties of such operator families are obtained. Relations between convoluted C-cosine functions and analytic convoluted C-semigroups, introduced and investigated in this paper are given through the convoluted version of the abstract Weierstrass formula which is also proved in the paper. Ultradistribution and hyperfunction sines are connected with analytic convoluted semigroups and ultradistribution semigroups. Several examples of operators generating convoluted cosine functions, (analytic) convoluted semigroups as well as hyperfunction and ultradistribution sines illustrate the abstract approach of the authors. As an application, it is proved that the polyharmonic operator Δn², n∈N, acting on L²[0,π] with appropriate boundary conditions, generates an exponentially bounded Kn-convoluted cosine function, and consequently, an exponentially bounded analytic Kn+1-convoluted semigroup of angle , for suitable exponentially bounded kernels Kn and Kn+1.