Dissipativity of the delay semigroup

Under mild conditions a delay semigroup can be transformed into a (generalized) contraction semigroup by modifying the inner product on the (Hilbert) state space into an equivalent inner product. Applications to stability of differential equations with delay and stochastic differential equations with delay are given as examples.     

The expansion of a semigroup and a Riesz basis criterion

Problems on the expansion of a semigroup and a criterion for being a Riesz basis are discussed in the present paper. Suppose that A is the generator of a C₀ semigroup on a Hilbert space and σ(A)=σ₁(A)∪σ₂(A) with σ₂(A) is consisted of isolated eigenvalues distributed in a vertical strip. It is proved that if σ₂(A) is separated and for each λ∈σ₂(A), the dimension of its root subspace is uniformly bounded, then the generalized eigenvectors associated with σ₂(A) form an L-basis. Under different conditions on the Riesz projection, the expansion of a semigroup is studied. In particular, a simple criterion for the generalized eigenvectors forming a Riesz basis is given. As an application, a heat exchanger problem with boundary feedback is investigated. It is proved that the heat exchanger system is a Riesz system in a suitable state Hilbert space.     

Stabilization through viscoelastic boundary damping: a semigroup approach

The undamped linear wave equation on a bounded domain in ℝ ⁿ with C ² boundary is considered. The interaction of the interior waves and the viscoelastic boundary material is modeled by convolution boundary conditions. It is assumed that the convolution kernel is integrable and completely monotonic. The main result is that the derivatives of all solutions tend to zero. The proof is given by an application of the Arendt-Batty-Lyubic-Vu Theorem. To this end, the model is reformulated as an abstract first order Cauchy problem in an appropriate Hilbert space, including the memory of the boundary as a state component. It is shown that the differential operator of the Cauchy problem is the generator of a contraction semigroup on the state space by establishing the range condition for the Lumer-Phillips Theorem using a generalized Lax-Milgram argument and Fredholm’s alternative. Furthermore, it is shown that neither the generator nor its adjoint have purely imaginary eigenvalues.     

Cellularity of twisted semigroup algebras

In this paper, the cellularity of twisted semigroup algebras over an integral domain is investigated by introducing the concept of cellular twisted semigroup algebras of type JH. Partition algebras, Brauer algebras and Temperley-Lieb algebras all are examples of cellular twisted semigroup algebras of type JH. Our main result shows that the twisted semigroup algebra of a regular semigroup is cellular of type JH with respect to an involution on the twisted semigroup algebra if and only if the twisted group algebras of certain maximal subgroups are cellular algebras. Here we do not assume that the involution of the twisted semigroup algebra induces an involution of the semigroup itself. Moreover, for a twisted semigroup algebra, we do not require that the twisting decomposes essentially into a constant part and an invertible part, or takes values in the group of units in the ground ring. Note that trivially twisted semigroup algebras are the usual semigroup algebras. So, our results extend not only a recent result of East, but also some results of Wilcox.     

Decompositions of a Krein space in regular subspaces invariant under a uniformly bounded C0-semigroup of bi-contractions

We give necessary and sufficient conditions under which a C₀-semigroup of bi-contractions on a Krein space is similar to a semigroup of contractions on a Hilbert space. Under these and additional conditions we obtain direct sum decompositions of the Krein space into invariant regular subspaces and we describe the behavior of the semigroup on each of these summands. In the last section we give sufficient conditions for the co-generator of the semigroup to be power bounded.     

Approximation of the semigroup generated by the Robin Laplacian in terms of the Gaussian semigroup

We consider the Laplacian ΔR subject to Robin boundary conditions on the space , where Ω is a smooth, bounded, open subset of RN. It is known that ΔR generates an analytic contraction semigroup. We show how this semigroup can be obtained from the Gaussian semigroup on C₀(RN) via a Trotter formula. As the main ingredient, we construct a positive, contractive, linear extension operator Eβ from to C₀(RN) which maps an operator core for ΔR into the domain of the generator of the Gaussian semigroup.     

On the structure of a semigroup of operators with finite-dimensional ranges

In the present paper, we describe the structure of a strongly continuous operator semigroup T(t) (where T: ?₊ → End X and X is a complex Banach space) for which ImT(t) is a finite-dimensional space for all t > 0. It is proved that such a semigroup is always the direct sum of a zero semigroup and a semigroup acting in a finite-dimensional space. As examples of applications, we discuss differential equations containing linear relations, orbits of a special form, and the possibility of embedding an operator in a C ₀-semigroup.     

Asymptotic convergence of nonlinear contraction semigroups in Hilbert space

Let S be a contraction semigroup on a closed convex subset C of a Hilbert space. If the generator of S satisfies a strengthened monotonicity condition then the weak lim_{t → ∞}S(t)x exists for all x in C. As one consequence, the method of steepest descent converges weakly for convex functions in Hilbert space; and it converges strongly for even convex functions.     

The non-nilpotent graph of a semigroup

We associate a graph ${\mathcal{N}}_{S}$ with a semigroup S (called the upper non-nilpotent graph of S). The vertices of this graph are the elements of S and two vertices are adjacent if they generate a semigroup that is not nilpotent (in the sense of Malcev). In case S is a group this graph has been introduced by A. Abdollahi and M.?Zarrin and some remarkable properties have been proved. The aim of this paper is to study this graph (and some related graphs, such as the non-commuting graph) and to discover the algebraic structure of S determined by the associated graph. It is shown that if a finite semigroup S has empty upper non-nilpotent graph then S is positively Engel. On the other hand, a semigroup has a complete upper non-nilpotent graph if and only if it is a completely simple semigroup that is a band. One of the main results states that if all connected ${\mathcal{N}}_{S}$ -components of a semigroup S are complete (with at least two elements) then S is a band that is a semilattice of its connected components and, moreover, S is an iterated total ideal extension of its connected components. We also show that some graphs, such as a cycle C _n on n vertices (with n??5), are not the upper non-nilpotent graph of a semigroup. Also, there is precisely one graph on 4 vertices that is not the upper non-nilpotent graph of a semigroup with 4 elements. This work also is a continuation of earlier work by Okni??ski, Riley and the first named author on (Malcev) nilpotent semigroups.     

Linearizability of non-expansive semigroup actions on metric spaces

We show that a non-expansive action of a topological semigroup S on a metric space X is linearizable iff its orbits are bounded. The crucial point here is to prove that X can be extended by adding a fixed point of S, thus allowing application of a semigroup version of the Arens-Eells linearization, iff the orbits of S in X are bounded.     

Every completely polynomially bounded operator is similar to a contraction

It is proved that a bounded operator on a Hilbert space is similar to a contraction if and only if it is completely polynomially bounded. This gives a partial answer to Problem 6 of Halmos (Bull. Amer. Math. Soc.76 (1970). 877–933). The set of completely bounded maps between C¹-algebras is studied to obtain some structure, representation, and extension theorems for this class of maps. These allow a characterization of the completely bounded representations, on a Hilbert space, of any subalgebra of a C¹-algebra to be obtained. The result in the title follows by applying this characterization to the disk algebra.     

Asymptotic behavior of nonlinear contraction semigroups

The asymptotic behavior of weak solutions of the equation $\text{{u(t) + Au(t) ? ?(t)}$ (A maximal monotone in Hilbert space) is determined via a characterization of ω-limit sets of the contraction semigroup generated by ?A.     

Construction of a unique self-adjoint generator for a symmetric local semigroup

A definition is given of a symmetric local semigroup of (unbounded) operators P(t) (0 ? t ? T for some T > 0) on a Hilbert space $\text{H}$, such that P(t) is eventually densely defined as t → 0. It is shown that there exists a unique (unbounded below) self-adjoint operator H on $\text{H}$ such that P(t) is a restriction of e^?tH. As an application it is proven that H₀ + V is essentially self-adjoint, where e^?tH₀ is an L^p-contractive semigroup and V is multiplication by a real measurable function such that V ∈ L^{2 + ε} and e^?δV ∈ L¹ for some ε, δ > 0.     

Stabilizability of two classes of contraction semigroups

This paper studies the strong stabilizability of two classes of Hilbert space contraction semigroups: (i) strict contraction semigroups, which include those with strictly dissipative generators; and (ii) isometric or unitary semigroups. The former class is already weakly stable, while the latter is not strongly stable over the whole space. Our tool is the functional model of Hilbert space contractions; hence, strong stability of the semigroup is studied via stability of its cogenerator. It is shown that a strict contraction semigroup is, in general, not strongly stabilized by the feedback –B*, while an isometric or a unitary semigroup is strongly stabilized by the same feedback, providedB is not compact.     

Two-point inequalities,the Hermite semigroup and the Gauss-Weierstrass semigroup

Let e^?zH, Re z ? 0, be the Hermite semigroup on R with Gauss measure μ. Necessary and sufficient conditions for e^?zH to be a bounded map from L^p(μ) into L^q(μ), 1 ? p, q ? ∞, are found and in many cases it is proved that e^?zH: L^p(μ) → L^q(μ) is in fact a contraction. Furthermore, these results and a formula relating the Hermite semigroup with the Gauss-Weierstrass semigroup e^zΔ enable one to calculate the precise norm of e^zΔ:L^p(dx) → L^q(dx) in a large number of cases.     

A groupoid approach to discrete inverse semigroup algebras

Let K be a commutative ring with unit and S an inverse semigroup. We show that the semigroup algebra KS can be described as a convolution algebra of functions on the universal étale groupoid associated to S by Paterson. This result is a simultaneous generalization of the author's earlier work on finite inverse semigroups and Paterson's theorem for the universal C^∗-algebra. It provides a convenient topological framework for understanding the structure of KS, including the center and when it has a unit. In this theory, the role of Gelfand duality is replaced by Stone duality.Using this approach we construct the finite dimensional irreducible representations of an inverse semigroup over an arbitrary field as induced representations from associated groups, generalizing the case of an inverse semigroup with finitely many idempotents. More generally, we describe the irreducible representations of an inverse semigroup S that can be induced from associated groups as precisely those satisfying a certain “finiteness condition.” This “finiteness condition” is satisfied, for instance, by all representations of an inverse semigroup whose image contains a primitive idempotent.     

The convergence of Lax-Oleinik semigroup for time-periodic Lagrangian

In this article,the convergence of so-called Lax-Oleinik semigroup is studied for time-periodic Lagrangian systems when the degree of freedom is greater than 2.Under certain conditions,we show that the Lax-Oleinik semigroup converges if the rotation vector is completely irrational.Removing such conditions,we will give another kind of convergence of the sequence Fc((x,s),(x′,s′+Tn)),the convergence of which is closely related to the Lax-Oleinik semigroup.     

Hypocoercivity for Kolmogorov backward evolution equations and applications

In this article we extend the modern, powerful and simple abstract Hilbert space strategy for proving hypocoercivity that has been developed originally by Dolbeault, Mouhot and Schmeiser in [16]. As well-known, hypocoercivity methods imply an exponential decay to equilibrium with explicit computable rate of convergence. Our extension is now made for studying the long-time behavior of some strongly continuous semigroup generated by a (degenerate) Kolmogorov backward operator L. Additionally, we introduce several domain issues into the framework. Necessary conditions for proving hypocoercivity need then only to be verified on some fixed operator core of L. Furthermore, the setting is also suitable for covering existence and construction problems as required in many applications. The methods are applicable to various, different, Kolmogorov backward evolution problems. As a main part, we apply the extended framework to the (degenerate) spherical velocity Langevin equation. This equation e.g. also appears in applied mathematics as the so-called fiber lay-down process. For the construction of the strongly continuous contraction semigroup we make use of modern hypoellipticity tools and perturbation theory.     

Dissipators for symmetric quasi-free dynamical semigroups on the CAR algebra

Let L be a negative self-adjoint bounded operator on a Hilbert space $\text{H}$, and p a projection on $\text{H}$ with pLp trace class, and let {T_t: t ? 0} be the extension of {e^tL: t ? 0} to a strongly continuous semigroup of completely positive quasi-free unital maps of Fock type on the fermion algebra A$\text{H}$ built over $\text{H}$. Then it is shown that there exists a strongly continuous self-adjoint contraction semigroup {G_t: t ? 0} on the Hilbert space of the GNS decomposition of the quasi-free state gw_p such that in the representation of that state: T_t ? G_t(·)G_t, t ?0.     

Every finite semigroup is embeddable in a finite relatively free semigroup

The title result is proved by a Murskii-type embedding.Results on some related questions are also obtained. For instance, it is shown that every finitely generated semigroup satisfying an identity ξ^d=ξ^2d is embeddable in a relatively free semigroup satisfying such an identity, generally with a larger d; but that an uncountable semigroup may satisfy such an identity without being embeddable in any relatively free semigroup.It follows from known results that every finite group is embeddable in a finite relatively free group. It is deduced from this and the proof of the title result that a finite monoid S is embeddable by a monoid homomorphism in a finite (or arbitrary) relatively free monoid if and only if its group of invertible elements is either {e} or all of S.