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.     

On some classes of infinitely differentiable operator semigroups

Given a linear relation (a multivalued linear operator), we construct an infinitely differentiable operator semigroup and study its properties.     

On perturbations preserving the immediate norm continuity of semigroups

We show that the Desch-Schappacher perturbation and the Miyadera-Voigt perturbation of an immediately norm continuous semigroup are immediately norm continuous. We also show that a perturbation theorem of C. Batty, C. Kaiser and L. Weis based on a generation theorem of A.M. Gomilko, D.-X. Feng and D.-H. Shi also preserves the immediate norm continuity of semigroups. The novelty of these results is that, contrary to the numerous related results, we obtain the immediate norm continuity of the perturbed semigroup without additional assumptions.     

Nonlinear perturbations of positive semigroups

Communicated by J. A. Goldstein     

On unbounded perturbations of semigroups: Compactness and norm continuity

Let A and B be the generators of strongly continuous semigroups (S(t)) _{t \geq 0} and (T(t)) _{t \geq 0} , respectively. Denote by Δ (t) = T(t) -S(t) . We show that if Δ(t) is norm continuous for t>0 and R(λ,B)-R(λ,A) is compact for λ ∈ ρ(A)\cap ρ(B) , then Δ(t) is compact. The converse is true if the perturbing operator is of Miyadera-Voigt-type. A characterization of norm continuity of Δ(t) in terms of the resolvents of the generators is given in Hilbert spaces.     

On the existence of invariant tori in nearly-integrable Hamiltonian systems with finitely differentiable perturbations

We prove the existence of invariant tori in Hamiltonian systems, which are analytic and integrable except a 2n-times continuously differentiable perturbation (n denotes the number of the degrees of freedom), provided that the moduli of continuity of the 2n-th partial derivatives of the perturbation satisfy a condition of finiteness (condition on an integral), which is more general than a Hölder condition. So far the existence of invariant tori could be proven only under the condition that the 2n-th partial derivatives of the perturbation are Hölder continuous.     

Form-bounded perturbations of generators of sub-Markovian semigroups

We develop perturbation theory of generators of sub-markovian semigroups by relatively form-bounded perturbations. The L ^p-smoothing properties of semigroups and the uniqueness problem are considered. Applications to operators of mathematical physics are given.     

Norm-ideal perturbations of one-parameter semigroups and applications

The notion of equivalence classes of generators of one-parameter semigroups based on the convergence of the Dyson expansion can be traced back to the seminal work of Hille and Phillips, who in Chapter XIII of the 1957 edition of their Functional Analysis monograph, developed the theory in minute detail. Following their approach of regarding the Dyson expansion as a central object, in the first part of this paper we examine a general framework for perturbation of generators relative to the Schatten-von Neumann ideals on Hilbert spaces. This allows us to develop a graded family of equivalence relations on generators, which refine the classical notion and provide stronger-than-expected properties of convergence for the tail of the perturbation series. We then show how this framework realises in the context of non-self-adjoint Schrödinger operators.     

Kernel estimates for perturbations of positive semigroups

Given a positive C⁰-semigroup T⁰ on L₂(Ω, m) with a kernel k⁰, where (Ω, m) is a σ-finite measure space, we study a suitably perturbed semigroup T and prove existence of a kernel k for T and an estimate of the k in terms of k⁰. In this way we extend a heat kernel estimate proven by Barlow, Grigor’yan and Kumagai [4] for Dirichlet forms perturbed by jump processes. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Positive semigroups and perturbations of boundary conditions

Positivity - We present a generation theorem for positive semigroups on an $$L^1$$ space. It provides sufficient conditions for the existence of positive and integrable solutions of...     

Rank-1 perturbations of cosine functions and semigroups

Let A be the generator of a cosine function on a Banach space X. In many cases, for example if X is a UMD-space, A+B generates a cosine function for each B∈L(D((ω−A)^1/2),X). If A is unbounded and , then we show that there exists a rank-1 operator B∈L(D(γ(ω−A)),X) such that A+B does not generate a cosine function. The proof depends on a modification of a Baire argument due to Desch and Schappacher. It also allows us to prove the following. If A+B generates a distribution semigroup for each operator B∈L(D(A),X) of rank-1, then A generates a holomorphic C₀-semigroup. If A+B generates a C₀-semigroup for each operator B∈L(D(γ(ω−A)),X) of rank-1 where 0<γ<1, then the semigroup T generated by A is differentiable and ‖T^′(t)‖=O(t−α) as t↓0 for any α>1/γ. This is an approximate converse of a perturbation theorem for this class of semigroups.     

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.     

Finite rank,relatively bounded perturbations of semigroups generators

Summary This paper is motivated by, and ultimately directed to, boundary feedback partial differential equations of both parabolic and hyperbolic type, defined on a bounded domain. It is written, however, in abstract form. It centers on the (feedback) operator A_F=A+P; A the infinitesimal generator of a s.c. semigroup on H; P an Abounded, one dimensional range operator (typically nondissipative), so that P=(A·, a)b, for a, b H. While Part I studied the question of generation of a s.c. semigroup on H by A_F and lack thereof, the present Part II focuses on the following topics: (i) spectrum assignment of A_F, given A and a H, via a suitable vector b H; alternatively, given A, via a suitable pair of vectors a, b H; (ii) spectrality of A_F—and lack thereof—when A is assumed spectral (constructive counterexamples include the case where P is bounded but the eigenvalues of A have zero gap, as well as the case where P is genuinely Abounded). The main result gives a set of sufficient conditions on the eigenvalues {_n} of A, the given vector a H and a given suitable sequence {_n} of nonzero complex numbers, which guarantee the existence of a suitable vector b H such that A_F possesses the following two desirable properties: (i) the eigenvalues of A_F are precisely equal to _n+_n; (ii) the corresponding eigenvectors of A_F form a Riesz basis (a fortiori, A_F is spectral). While finitely many _ns can be preassigned arbitrarily, it must be however that _n 0 « sufficiently fast ». Applications include various types of boundary feedback stabilization problems for both parabolic and hyperbolic partial differential equations. An illustration to the damped wave equation is also included.Research partially supported by Air Force Office of Scientific Research under Grant AFOSR-84-0365.