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.     

Bands of k-Archimedean semigroups

In this paper we define the radical ϱ _k (k∈Z ⁺) of a relation ϱ on an arbitrary semigroup. Also, we define various types of k-regularity of semigroups and various types of k-Archimedness of semigroups. Using these notions we describe the structure of semigroups in which ρ _k is a band (semilattice) congruence for some Green’s relation.     

A note on approximation of semigroups of contractions on Hilbert spaces

We show that every contractive C ₀-semigroup on a separable, infinite-dimensional Hilbert space X can be approximated by unitary C ₀-groups in the weak operator topology uniformly on compact subsets of ℝ₊. As a consequence we get a new characterization of a bounded H ^∞-calculus for the negatives of generators of bounded holomorphic semigroups. Applications of our results to the study of a topological structure of the set of (almost) weakly stable contractive C ₀-semigroups on X are also discussed. The author was partially supported by the Marie Curie “Transfer of Knowledge” programme, project “TODEQ”, and by a MNiSzW grant Nr. N201384834.     

Difference estimates for continuous and discrete operator semigroups

For suitable bounded operator semigroups (e ^tA ) _t≥0 in a Banach space, we characterize the estimate ‖Ae ^tA ‖≤c/F(t) for large t, where F is a function satisfying a sublinear growth condition. The characterizations are by holomorphy estimates on the semigroup, and by estimates on powers of the resolvent. We give similar characterizations of the difference estimate ‖T ⁿ −T ⁿ⁺¹‖≤c/F(n) for a power-bounded linear operator T, when F(n) grows faster than n ^1/2 for large n.     

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.     

Chaotic Tensor Product Semigroups

Suppose {G₁(t)}_{t ≥ 0} and {G₂(t)_{t ≥ 0} be two semigroups on an infinite dimensional separable reflexive Banach space X. In this paper we give sufficient conditions for tensor product semigroup G(t): X → G₂(t)X G₁(t) to become chaotic in L with the strong operator topology and chaotic in the ideal of compact operators on X with the norm operator topology.     

Approximating a Feller Semigroup by Using the Yosida Approximation of the Symbol of its Generator

We show that if the pseudodifferential operator −q(x,D) generates a Feller semigroup (T_t)_t≥0 then the Feller semigroups (T_t^(v))_t≥0 generated by the pseudodifferential operators with symbol will converge strongly to (T_t)_t≥0 as ν →∞.     

Point interactions inL p

An interpretation is given to point interactions of the form −Δ+d inL ^p (ℝ ^N ), where Δ is the Laplacian operator andd is a pseudopotential related to the ‘Dirac measure at 0', depending on the dimension. They are described as extensions of −Δ, defined on the space {u∈C ₀ ^∞ (ℝ ^N )|u(0)=0} that are negative generators of analytic semigroups. This is done forN=1,2 and 1<p<∞ and forN=3 and 3/2<p<3.     

The Maximal Operator Associated to a Nonsymmetric Ornstein–Uhlenbeck Semigroup

Let (ℋ _t ) _t≥0 be the Ornstein–Uhlenbeck semigroup on ℝ ^d with covariance matrix I and drift matrix λ(R−I), where λ>0 and R is a skew-adjoint matrix, and denote by γ _∞ the invariant measure for (ℋ _t ) _t≥0. Semigroups of this form are the basic building blocks of Ornstein–Uhlenbeck semigroups which are normal on L ²(γ _∞). We prove that if the matrix R generates a one-parameter group of periodic rotations, then the maximal operator ℋ_* f(x)=sup  _t≥o |ℋ _t f(x)| is of weak type 1 with respect to the invariant measure γ _∞. We also prove that the maximal operator associated to an arbitrary normal Ornstein–Uhlenbeck semigroup is bounded on L ^p (γ _∞) if and only if 1<p≤∞.      

Mean ergodic theorems for bi-continuous semigroups

In this paper we study the main properties of the Cesàro means of bi-continuous semigroups, introduced and studied by Kühnemund (Semigroup Forum 67:205–225, 2003). We also give some applications to Feller semigroups generated by second-order elliptic differential operators with unbounded coefficients in C _b (ℝ ^N ) and to evolution operators associated with nonautonomous second-order differential operators in C _b (ℝ ^N ) with time-periodic coefficients.     

On numerical experiments with symmetric semigroups generated by three elements and their generalization

We give a simple explanation of numerical experiments of V. Arnold with two sequences of symmetric numerical semigroups, S(4,6+4k,87−4k) and S(9,3+9k,85−9k) generated by three elements. We present a generalization of these sequences by numerical semigroups S(r₁²,r₁r₂+r₁²k,r₃-r₁²k)\mathsf{S}(r_{1}^{2},r_{1}r_{2}+r_{1}^{2}k,r_{3}-r_{1}^{2}k), k∈ℤ, r ₁,r ₂,r ₃∈ℤ⁺, r ₁≥2 and gcd(r ₁,r ₂)=gcd(r ₁,r ₃)=1, and calculate their universal Frobenius number Φ(r ₁,r ₂,r ₃) for the wide range of k providing semigroups be symmetric. We show that this type of semigroups admit also nonsymmetric representatives. We describe the reduction of the minimal generating sets of these semigroups up to {r₁²,r₃-r₁²k}\{r_{1}^{2},r_{3}-r_{1}^{2}k\} for sporadic values of k and find these values by solving the quadratic Diophantine equation.     

Integrated semigroups,C-semigroups and the abstract Cauchy problem

LetA be a linear operator on a Banach space. We consider when the following holds. (*)u′(t,x)=A(u(t,x)) (t≥0),u(0,x)=x, has a unique solution, for allx in the domain ofA ⁿ⁺¹ . We discuss the relationship between (*), integrated semigroups, and C-semigroups. We use this to obtain new results about integrated semigroups and the abstract Cauchy problem. We give several examples where (*) may be easily shown using C-semigroups. Many of these examples may not be done directly using integrated semigroups.     

Chain decompositions of ordered semigroups

Characterizations of ordered semigroups which can be decomposed into (natural ordered) chains of ω -simple ordered semigroups are given, where ω -simple ordered semigroups are ξ _l (ξ _t ) -simple, left (t -) simple, L _n (H _n ) -simple, l (t )-archimedean and nil-extensions of left (t -) simple ordered semigroups, respectively. As a generalization of the theory of Clifford semigroups (without orders) to ordered semigroups, ordered semigroups which are semilattices of t -simple subsemigroups are characterized.     

The Heat Equation for the Generalized Hermite and the Generalized Landau Operators

We give a formula for the one-parameter strongly continuous semigroups ${e^{-tL^{\lambda}}}We give a formula for the one-parameter strongly continuous semigroups e^-tLl{e^{-tL^{\lambda}}} and e^{-t [(A)\tilde]}{e^{-t \tilde{A}}}, t > 0 generated by the generalized Hermite operator L^l, l ? R\{0}{L^{\lambda}, \lambda \in {\bf R}\backslash \{0\}} respectively by the generalized Landau operator ?. These formula are derived by means of pseudo-differential operators of the Weyl type, i.e. Weyl transforms, Fourier-Wigner transforms and Wigner transforms of some orthonormal basis for L ²(R ²ⁿ ) which consist of the eigenfunctions of the generalized Hermite operator and of the generalized Landau operator. Applications to an L ² estimate for the solutions of initial value problems for the heat equations governed by L ^λ respectively ?, in terms of L ^p norm, 1 ≤ p ≤ ∞ of the initial data are given.     

On the Carleson Measure Criterion in Linear Systems Theory

In Ho and Russell (SIAM J Control Optim 21(4):614–640, 1983), and Weiss (Syst Control Lett 10(1): 79–82, 1988), a Carleson measure criterion for admissibility of one-dimensional input elements with respect to diagonal semigroups is given. We extend their results from the Hilbert space situation (L ²-admissibility on the state space ℓ ₂) to the more general situation of L ^p -admissibility on the state space ℓ _q . For analytic diagonal semigroups we present a new result that does not rely on Laplace transform methods. A comparison of both criteria leads to a result on L ^p -admissibility for reciprocal systems in the sense of Curtain (Syst Control Lett 49(2):81–89, 2003).     

Uniqueness of Embedding into a Gaussian Semigroup and a Poisson Semigroup with Determinate Jump Law on a Simply Connected Nilpotent Lie Group

Let {μ _t ⁽ⁱ⁾} _t≥0 (i=1,2) be continuous convolution semigroups (c.c.s.) on a simply connected nilpotent Lie group G. Suppose that μ ₁⁽¹⁾=μ ₁⁽²⁾. Assume furthermore that one of the following two conditions holds:

On the invertibility of the operator <Emphasis Type="Italic">d/dt</Emphasis> + <Emphasis Type="Italic">A</Emphasis> in certain functional spaces

We prove that the operator d/dt + A constructed on the basis of a sectorial operator A with spectrum in the right half-plane of ℂ is continuously invertible in the Sobolev spaces W _p ¹ (ℝ, D _α), α ≥ 0. Here, D _α is the domain of definition of the operator A ^α and the norm in D _α is the norm of the graph of A ^α. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1020–1025, August, 2007.     

On generators and relations for unions of semigroups

Finite generation and presentability of general unions of semigroups, as well as of bands of semigroups, bands of monoids, semilattices of semigroups and strong semilattices of semigroups, are investigated. For instance, it is proved that a band Y of monoids S _α (α∈ Y ) is finitely generated/presented if and only if Y is finite and all S _α are finitely generated/presented. By way of contrast, an example is exhibited of a finitely generated semigroup which is not finitely presented, but which is a disjoint union of two finitely presented subsemigroups. January 21, 2000     

Regular Couplings of Dissipative and Anti-Dissipative Unbounded Operators,Asymptotics of the Corresponding Non-Dissipative Processes and the Scattering Theory

In this paper a triangular model of a class of unbounded non-selfadjoint K ^r-operators A presented as a coupling of dissipative and anti-dissipative operators in a Hilbert space with real absolutely continuous spectra and with different domains of A and A ^* is considered. The asymptotic behaviour of the corresponding non-dissipative processes T_tf = e^itAf, generated from the semigroups T_t with generators iA, as t → ± ∞ are obtained. The strong wave operators, the scattering operator for the couple (A^*, A) and the similarity of A and the operator of multiplication by the independent variable are obtained explicitly. The considerations are based on the triangular models and characteristic functions of A. Kuzhel for unbounded operators and the limit values of the multiplicative integrals, describing the characteristic function of the considered model. Partially supported by Grant MM-1403/04 of MESC and by Scientific Research Grant 27/25.02.2005 of Shumen University.