Perturbations of positive semigroups on AM-spaces

We consider positive perturbations of positive semigroups on AM-spaces and prove a result which is the dual counterpart of a famous perturbation result of Desch in AL-spaces. As an application we present unbounded perturbations of the shift semigroup.     

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)     

Perturbations of differentiable semigroups

If (A, D(A)) generates a C ₀-semigroup T on a Banach space X and then (A + B, D(A)) is also the generator of a C ₀-semigroup, S _B . There are easy examples to show that if T is eventually differentiable then S _B need not be eventually differentiable. In 1995 an example was constructed to show that if T is immediately differentiable then S _B need not be immediately differentiable. In this paper we establish necessary and sufficient conditions on the generator (A, D(A)) of T which ensure that eventual or immediate differentiability of T is inherited by S _B for all . We are therefore able to give a characterization of the immediately and eventually differentiable C ₀-semigroups for which differentiability is a stable property under bounded perturbations of the generator. We also prove a characterization of the C ₀-semigroups for which the norm of the resolvent of the generator decays on vertical lines and a new characterization of the Crandall-Pazy class of semigroups. We are grateful to Charles Batty and Tom Ransford for helpful discussions and to the referee for their constructive comments.     

Perturbations and approximations of integrated semigroups

The purpose of this paper is to establish some basic results about perturbations and approximation of integrated semigroups. The Project Supported by National Science Fundation of China.     

Diagonals of positive semigroups

LetE be a Dedekind complete complex Banach lattice and letD denote the diagonal projection from the spaceL _r (E) onto the centerZ(E) ofE. Let {T(t)} _t0 be a positive strongly continuous semigroup of linear operators with generatorA. The first main result is that if the spectral bounds(A) equals to zero, then the functionD(T(t)) is a center valuedp-function. The second main result is that if for >0 the diagonalD(R(, A)) of the resolvent operatorR(, A) is strictly positive, then (D(R(, A))) ^–1 is a center valued Bernstein function. As an application of these results it follows that the order limit lim₀D(R(,A)) exists inZ(E) and equals the order limit lim _m D((R(, A)) ^m ) for any >0.     

Perturbations of eventually differentiable and eventually norm-continuous semigroups on Banach spaces

In this paper we discuss perturbations of eventually differentiable and eventually norm-continuous semigroups on a Banach space. Two kinds of new perturbation theorems are obtained.     

Factoring of positive definite functions on semigroups

No abstract. June 15,1999     

The generators of positive semigroups

We characterise the infinitesimal generators of norm continuous one-parameter semigroups of positive maps on certain ordered spaces, with special reference to C¹-algebras.     

Nonlinear perturbations of positive semigroups

Communicated by J. A. Goldstein     

Perturbations of finite networks and asymptotic periodicity of flow semigroups

We consider a transport process on a finite network and introduce a new special transformation of the underlying graph that is then used to yield an alternative proof of the main theorem on asymptotic periodicity of the flow semigroup proven in Kramar and Sikolya (Math. Z. 249:139–162, 2005). The perturbation technique we develop allows us to prove this theorem starting from a special case, thus greatly simplifying the proof. This technique is in Kunszenti-Kovács (Perturbations of networks and asymptotic periodicity of recurrent flows: the infinite case, Preprint, 2008) adapted to infinite networks, where it allows for significant generalisations of previous results.     

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.     

Smooth positive definite functions on some multiplicative semigroups

Smoothness of positive definite functions on multiplicative semigroups of the form ]−1, 1[ ^d ,d≥1, is characterized in terms of their representing measures. Some extension questions concerning functions of this type are discussed.