Let (U(t))t≥0 be a C0-semigroup of bounded linear operators on a Banach space X. In this paper, we establish that if, for some t0 > 0, U(t0) is a Fredholm (resp., semiFredholm) operator, then (U(t))t≥0 is a Fredholm (resp., semi-Fredholm) semigroup. Moreover, we give a necessary and sufficient condition guaranteeing that (U(t))t≥0 can be embedded in a C0-group on X. Also we study semigroups whi...

We study the number of solutions of the general semigroup equation in one variable, X α = X β , as well as of the system of equations X 2 = X, Y 2 = Y, XY = Y X in H ≀ T n , the wreath product of an arbitrary finite group H with the full transformation semigroup T n on n letters. For these solution numbers, we provide explicit exact formulae, as well as asymptotic estimates. Our results concern...

A CP -semigroup (or quantum dynamical semigroup) is a semigroup φ = {φt : t ≥ 0} of normal completely positive linear maps on B(H), H being a separable Hilbert space, which satisfies φt(1) = 1 for all t and is continuous in the natural sense. Let D be the natural domain of the generator L of φ, φt = exp tL. Since the maps φt need not be multiplicative D is typically an operator space, but not a...

The fractional order differential equation u′(t) = Au(t) + γD t Au(t) + f(t), t > 0, u(0) = a ∈ X is studied, whereA is an operator generating a strongly continuous one-parameter semigroup on a Banach space X , D t is the Riemann–Liouville fractional derivative of order α ∈ (0, 1), γ > 0 and f is an X-valued function. Equations of this type appear in the modeling of unidirectional viscoelastic ...

We study the existence and uniqueness of mild and classical solutions for a nonlinear impulsive evolution equation u′(t) = Au(t) + f(t, u(t)), 0 < t < T0, t = ti, u(0) = u0, ∆u(ti) = Ii(u(ti)), i = 1, 2, ..., 0 < t1 < t2 < ... < T0, in a Banach space X, where A is the generator of a strongly continuous semigroup, ∆u(ti) = u(t+i ) − u(ti ), and Ii’s are some operators. The impulsive conditions c...

Let Γ be a semigroup of d × d nonsingular integer matrices, and consider the action of Γ on the torus T. We assume throughout that the action is strongly irreducible: there is no subtorus invariant under a finite index subsemigroup of Γ. The strong irreducibility assumption in particular implies that Γ acts ergodically on T (equipped with the Lebesgue measure m). Therefore the Γ-orbit of Lebesg...

Let E be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. Let J {T t : t ≥ 0} be a family of uniformly asymptotically regular generalized asymptotically nonexpansive semigroup of E, with functions u, v : 0,∞ → 0,∞ . Let F : F J ∩t≥0F T t / ∅ and f : K → K be a weakly contractive map. For some positive real numbers λ and δ satisfying δ λ > 1, let G ...

Using a generalization of the semigroup theory of linear operators, we prove existence and uniqueness of S-asymptotically !-periodic mild solutions for a class of linear and semilinear fractional order differential equations of the form D +1 t u(t) + D t u(t)−Au(t) = f(t, u(t)), t > 0, 0 < ≤ ≤ 1, ≥ 0.

For a strongly continuous semigroup (T (t))t≥0 with generator A we introduce its critical spectrum σcrit(T (t)). This yields in an optimal way the spectral mapping theorem σ(T (t)) = e ∪ σcrit(T (t)) and improves classical stability results.