C-semigroups and strongly continuous semigroups

We show that, whenA generates aC-semigroup, then there existsY such that [M(C)] →Y →X, andA| _Y , the restriction ofA toY, generates a strongly continuous semigroup, where ↪ means “is continuously embedded in” and ‖x‖_[Im(C)]≡‖C ⁻¹ x‖. There also existsW such that [C(W)] →X →W, and an operatorB such thatA=B| _X andB generates a strongly continuous semigroup onW. If theC-semigroup is exponentially bounded, thenY andW may be chosen to be Banach spaces; in general,Y andW are Frechet spaces. If ρ(A) is nonempty, the converse is also true. We construct fractional powers of generators of boundedC-semigroups. We would like to thank R. Bürger for sending preprints, and the referee for pointing out reference [37]. This research was supported by an Ohio University Research Grant.     

Invariant subsets of strongly continuous semigroups

Let {P(t): t0} be a strongly continuous semigroup on a Banach space X and let |\| be a continuous norm on X such that |P(t)x|exp(t)|x|, XX, t0. Let C be a |\|-closed convex subset of X and suppose that for every x in D(A) there exists a sequence (x_n : n ) in D(A) with the following properties: lim|x–x_n|=0, lim|Ax–Ax_n|=0 and every x_n has a best approximation in C (with respect to |\|) which belongs to D(A). Then P(t)CC for all t0 if and only if, for every v in CD(A), the vector Av belongs to the |\|-closure of [0, ) (C-V).     

Stabilized approximations of strongly continuous semigroups

This paper introduces stabilization techniques for intrinsically unstable, high accuracy rational approximation methods for strongly continuous semigroup. The methods not only stabilize the approximations, but improve their speed of convergence by a magnitude of up to 1/2.     

Embedding operators into strongly continuous semigroups

We study linear operators T on Banach spaces for which there exists a C₀-semigroup (T(t))_t≥0 such that T = T(1). We present a necessary condition in terms of the spectral value 0 and give classes of examples for which such a C₀-semigroup does or does not exist. Received: 22 December 2008, Revised: 7 April 2009     

Stability of strongly continuous representations of abelian semigroups

Part of this work was carried out while the authors were visiting the University of Franche-Comté, Besan?on, and Hokkaido University, Sapporo, respectively     

A spectral mapping theorem for perturbed strongly continuous semigroups

The second author was supported by a F.P.I. grant from Spanish Ministerio de Educación y Ciencia     

Strong convergence theorems for strongly continuous semigroups of pseudocontractions

In this work, strongly continuous semigroups of pseudocontractions are studied based on an implicit iterative algorithm. Strong convergence theorems of fixed points are obtained in an arbitrary Banach space.     

Perturbations of positive strongly continuous semigroups on AM spaces

We give a definition for positivity on an extrapolation space for positive strongly continuous semigroups. We prove a perturbation result on AM spaces for positive semigroups by positive operators. Finally, we give an example for this kind of perturbations. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the spectral mapping theorem for perturbed strongly continuous semigroups

We consider a strongly continuous semigroup (T(t))_{t \geqq 0}(T(t))_{t \geqq 0} with generator A on a Banach space X, an A-bounded perturbation B, and the semigroup (S(t))_{t \geqq 0}(S(t))_{t \geqq 0} generated by A + B. Using the critical spectrum introduced recently, we improve existing spectral mapping theorems for the perturbed semigroup (S(t))_{t \geqq 0}(S(t))_{t \geqq 0} .     

Robust controller for systems with exponentially stable strongly continuous semigroups

The theory of robust controllers is extended to the case where we have boundary and/or distributed control and the system operator is an infinitesimal generator of a strongly continuous exponentially stable semigroup. An example on hyperbolic systems is presented.     

Perturbations of strongly continuous operator semigroups,and matrix Muckenhoupt weights

Let A and A ₀ be linear continuously invertible operators on a Hilbert space ? such that A ^?1 ? A ₀ ^?1 has finite rank. Assuming that σ(A ₀) = ? and that the operator semigroup V ₊(t) = exp{iA ₀ t}, t ≥ 0, is of class C ₀, we state criteria under which the semigroups U _±(t) = exp{±iAt}, t ≥ 0, are of class C ₀ as well. The analysis in the paper is based on functional models for nonself-adjoint operators and techniques of matrix Muckenhoupt weights.     

Implicit iterative algorithms for treating strongly continuous semigroups of Lipschitz pseudocontractions

In this work, theorems of weak convergence of an implicit iterative algorithm with errors for treating strongly continuous semigroups of Lipschitz pseudocontractions are established in the framework of real Banach spaces.     

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.     

A sharp result on the exponential operator-norm decay of a family of strongly continuous semigroups

Given a familyT _h (t) of strongly continuous semigroups on a Banach spaceX, we give a sharp result on the exponential decay inL(X) of the family, which is uniform in the parameterh. Sharpness is shown by means of counterexamples. This result generalizes a well-known and very useful criterion of stability of a single strongly continuous semigroup. The obtained generalization plays a critical role in a variety of applications, from approximation theory to regularization theory. Research partially supported by the National Science Foundation under Grant NSF-DMS-8902811-01.