On multiplicative perturbations of C 0 -groups and C 0 -cosine operator functions

Certain convolution operators of the form (K f) (t) = A∈t ₀ ^t L(t-s) f(s) ds , where A is the infinitesimal generator of either a C ₀ -group or a C ₀ -cosine family in a Banach space E , are considered. We obtain several lifting results guaranteeing that the continuity of K from L ^p to L ^q implies the continuity of K from L ^p to L ^∈ fty . These results are applied to the study of multiplicative perturbations of C ₀ -groups and C ₀ -cosine families in Banach spaces and to the study of the Maximal Regularly Property (MRP) in L ^p , 1 ≤ p ≤ +∈fty , for second-order Cauchy problem. It is proved that the MRP is equivalent to the boundedness of the infinitesimal generator. April 30, 1999     

A characterization of uniformly bounded cosine functions generators

Suported by DAAD, West Germany.     

A functional calculus for analytic generators ofC 0-groups

In [9] and [3] anF(S )-functional calculus for sectorial operators is constructed via the Dunford-Riesz integral. This calculus implicitely defines the well-known complex powers of such operators. Sectorial operators with bounded imaginary powers turn out to be of particular interest due to the remarkable Dore-Venni theorem. In [12] this theorem is proved via the theory of analytic generators ofC ₀-groups. These results suggest the existence ofF(S )-functional, calculi forC ₀-groups and their analytic generators. In this paper we show that such functional calculi indeed exsist, however the approach via the Dunford-Riesz integral is no longer viable. Therefore a different approach via an approximation argument is introduced. Existence and uniqueness theorems are given and we show how the functional calculi relate to known results. Examples illustrate the theory.     

C 0-groups with Polynomial Growth

We consider some conditions guaranteeing that an unbounded operator is a generator of some strongly continuous group satisfying a polynomial growth condition. The result is applied to a Friedrichs model given by Tf(x)=xf(x)+<f,ϕ>ψ (x) and considered as an operator in L ^p (R). Some sufficient condition on ϕ and ψ are found such that iT generates an example of a group with polynomial growth. April 30, 1999     

Spectral properties of generators of theC 0-groups over tensor products of Banach spaces

We study the properties of common spectral subspaces of the N generators of one-parameter (C₀)-groups on different Banach spaces. The method of study is based on the functional calculus in the convolution algebra L¹ (R^N). We establish a theorem on the equality of the common spectra of the generators and their restrictions to common spectral subspaces.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 40–47.     

Kato's equality and spectral decomposition for positive C0-groups

Let A be the generator of a C₀-semigroup {T(t); t0} defined on a Banach lattice E. It is shown that T(t) is a lattice homomorphism for all t>0 if and only if A satisfies <¦x¦, Ax>= (xD(A), x D(A)) (where q: EE is the evaluation mapping). This equality is used to obtain a spectral decomposition for generators of positive groups.     

Interpolation of cosine operator functions

Summary Using basic techniques from the theory of interpolation spaces equivalence theorems are established for the intermediate spaces between a given Banach space A and the domain D(^r) of the r-th power of the infinitesimal generator of a strongly continuous cosine operator function C. The results are applied to the study of second order evolution equations including regularity, order reduction and approximation by finite difference methods.     

Perturbation and comparison of cosine operator functions

LetA be a closed linear operator such that the abstract Cauchy problemu″(t)=Au(t), t∈R; u(0)=x, u′(0)=y, is well-posed. We present some multiplicative perturbation theorems which give conditions on an operatorC so that the abstract Cauchy problems for differential equationsu″(t)=ACu(t) andu″(t)=CAu(t) also are well-posed. Some new or known additive perturbation theorems and mixed-type perturbation theorems are deduced as corollaries. Applications to characterization of the infinitesimal comparison of two cosine operator functions are also discussed. Research supported in part by the National Science Council of Taiwan.     

Matrices of operators and regularized cosine functions

This paper is concerned with matrices of abstract differential operators which are parabolic in the sense of Shilov or correct in the sense of Petrovskij. We show that they generate regularized cosine functions with suitable regularizing operators under sharper conditions. The results then are applied to matrices of partial differential operators on many function spaces. Finally, the wellposedness of the associated second-order systems is discussed.     

On moments-preserving cosine families and semigroups in C[0, 1]

We use the newly developed Lord Kelvin’s method of images (Bobrowski in J Evol Equ 10(3):663–675, 2010; Semigroup Forum 81(3):435–445, 2010) to show existence of a unique cosine family generated by a restriction of the Laplace operator in C[0, 1] that preserves the first two moments. We characterize the domain of its generator by specifying its boundary conditions. Also, we show that it enjoys inherent symmetry properties, and in particular that it leaves the subspaces of odd and even functions invariant. Furthermore, we provide information on long-time behavior of the related semigroup.     

On a characteristic of cosine functions

This paper is concerned with the property of cosine function. It is proved that a family {T(t)} _t≥0 of strongly continuous linear operators is a cosine function on Banach space X if and only if T(0)=I and there holds $$\begin{aligned} \int_0^{t+s}T(\tau)d\tau=T(t)\int _0^s T(\tau)d\tau+\int_0^t T(\tau)d\tau T(s),\quad t,s \geq0, \end{aligned}$$ where all the integrals concerning operator valued functions are understood to be in the strong operator topology.     

Spectral properties of cosine operator functions

LetA be the generator of a cosine functionC _t ,t R in a Banach spaceX; we shall connect the existence and uniqueness of aT-periodic mild solution of the equationu = Au + f with the spectral property 1 (C _T ) and, in caseX is a Hilbert space, also with spectral properties ofA. This research was supported in part by DAAD, West Germany.     

Approximation of semigroups and cosine functions in spaces of periodic functions

In this article we furnish a representation of the solutions of some classes of first-order and second-order evolution problems as limit of iterates of classical sequences of approximating operators. The method is based on Trotter's theorem on the approximation of semigroups which is applied here also for the approximation of groups and cosine functions. We apply this method in spaces of continuous periodic functions and using some classical sequences of trigonometric polynomials.     

Rigid extensions of ℓ-groups of continuous functions

Let C(X, ℤ), C(X, ℂ) and C(X) denote the ℓ-groups of integer-valued, rational-valued and real-valued continuous functions on a topological space X, respectively. Characterizations are given for the extensions C(X, ℤ) ⩽ C(X, ℚ) ⩽ C(X) to be rigid, major, and dense.     

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.