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.     

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.     

On the spectrum of cosine operator functions

In this paper we characterize the spectrum of strongly continuous cosine functions, defined in a Hilbert space, in terms of properties of the infinitesimal generator.     

On operator cosine functions in UMD spaces

We study representations, group decomposition and the inversion of the Laplace transform for cosine operator functions in UMD spaces. February 28, 2000     

On bounded time-dependent perturbations of operator cosine functions

In this note it is shown that the Cauchy problem associated with the infinitesimal generatorA of a strongly continuous operator cosine function remains uniformly well-posed under bounded time-dependent perturbations ofA.     

Opial type inequalities for cosine and sine operator functions

Various L _p form Opial type inequalities are given for cosine and sine operator functions with applications.     

The probabilistic representation formulae for cosine operator functions

In this paper we obtained general representation formulae for strongly continuous cosine operator functions via probabilistic approach, which include Webb’s [1] and Shaw’s [2] formulae and some new one as special cases. We also give the quantitative estimations for the general formulae.     

Growth order and stability of semigroups and cosine operator functions

Equivalent conditions for a trajectory of a C₀-semigroup T(⋅) (resp. cosine function C(⋅)) of operators to have the growth order O(tα) or o(tα) are expressed in terms of Cesàro and Abel means of the norm of the trajectory. We then deduce characterizations of growth order and stability for T(⋅) and C(⋅). It is also shown that under some Tauberian condition the uniform boundedness (resp. strong convergence) of T(⋅) is equivalent to the uniform boundedness (resp. strong convergence) of its Abel means.     

STRONGLY CONTINUOUS INTEGRATED COSINE OPERATOR FUNCTIONS WITH GROWTH ω

For a continuous increasing function ω ： [0, ∞） → （0, ∞） of finite exponetial type, we establish a Hille-Yosida type theorem for strongly continuous integrated cosine operator functions with O（ω）. It includes the well-known polynomially bounded and exponentially bounded cases.     

Strong and uniform mean stability of cosine and sine operator functions

It is first observed that a uniformly bounded cosine operator function C() and the associated sine function S() are totally non-stable. Then, using a zero-one law for the Abel limit of a closed linear operator, we prove some results concerning strong mean stability and uniform mean stability of C(). Among them are: (1) C() is strongly (C,1)-mean stable (or (C,2)-mean stable, or Abel-mean stable) if and only if 0ρ(A)σ_c(A); (2) C() is uniformly (C,2)-mean stable if and only if S() is uniformly (C,1)-mean stable, if and only if , if and only if , if and only if C() is uniformly Abel-mean stable, if and only if S() is uniformly Abel-mean stable, if and only if 0ρ(A).     

Compactness properties in the theory of the cosine operator functions

Translated from Matematicheskie Zametki, Vol. 51, No. 5, pp. 151–153, May, 1992.     

Semigroups of operators,cosine operator functions,and linear differential equations

This survey presents a systematic exposition of the elements of the theory of operator semigroups (OS's) in Banach space from Hille-Yosida to the end of 1989. There is a parallel exposition of the theory of cosine operator functions (COF's). The paper contains the following divisions: Linear differential equations in Banach space, reduction of the Cauchy problem for second order equations to the Cauchy problem for first order equations, one-parameter OS's and COF's. differentiable OS's. analytic OS's. Fredholm OS's. positive OS's. stable OS's. spectral properties of OS's and COF's. compactness properties of OS's and COF's. uniformly continuous OS's and COF's. almost periodic OS's and COF's. uniformly bounded OS's and COF's. the theory of perturbations for OS's and COF's. ad joint OS's and COF's.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 28, pp. 87–202, 1990.     

Poincaré like inequalities for semigroups, cosine and sine operator functions

Here we present Poincaré type general L _p inequalities regarding semigroups, cosine and sine operator functions.     

Some applications of Fejer's theorem to operator cosine functions in Banach spaces

We characterize spectral properties of operator cosine functions in Banach spaces in terms of the Césaro summability of two series associated to the resolvent of the corresponding infinitesimal generator.

     

The mean ergodic theorem for cosine operator functions with optimal and non-optimal rates

Dedicated to Professor Károly Tandori on the occasion of his 70 th birthday, in great respect     

Hypercyclicity and mixing for cosine operator functions generated by second order partial differential operators

For Ω⊂Rd open, we characterize when cosine operator functions generated by second order partial differential operators on Lp(Ω,μ) and C0,ρ(Ω), respectively, are hypercyclic and prove that this happens if and only if they are weakly mixing. In the case of d=1 we give an easy to check characterization of when this happens. Moreover, mixing of these cosine operator functions is also characterized.