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.     

Über Die Konvergenz Operatorwertiger Cosinusfunktionen mit Gestörtem Infinitesimalen Erzeuger

On convergence of operator cosine functions with perturbed infinitesimal generator. The question under what kind of perturbations a closed linear operatorA remains of the class of infinitesimal generators of operator cosine functions seems to be a rather difficult one and is unsolved in general. In this note we give bounds for the perturbation of operator cosine functions caused byA-bounded perturbationsT ofA under the assumption thatT + A is also a generator.

     

On spectral variations under bounded real matrix perturbations

Summary In this paper we investigate the set of eigenvalues of a perturbed matrix {ie509-1} whereA is given and ^{n × n}, ||< is arbitrary. We determine a lower bound for thisspectral value set which is exact for normal matricesA with well separated eigenvalues. We also investigate the behaviour of the spectral value set under similarity transformations. The results are then applied tostability radii which measure the distance of a matrixA from the set of matrices having at least one eigenvalue in a given closed instability domain _b.     

On some operator monotone functions

We try to find a continuous functionu defined on a real right half-line with the range (0, ) such thatu ^–1 is operator monotone. We then look for another functionv such thatv(u ^–1) is operator monotone, namely,u(A)u(B) impliesv(A)v(B) for self-adjoint operatorsA andB.     

On an alternative cauchy equation in two unknown functions. Some classes of solutions

Summary In this paper we consider the alternative Cauchy functional equationg(xy) g(x)g(y) impliesf(xy) = f(x)f(y) wheref, g are functions from a topological group (X, ·) into a group (S,·). First we prove that, ifS is a Hausdorff topological group andX satisfies some weak additional hypotheses, then (f, g) is a continuous solution if and only if eitherf org is a homomorphism. Then we describe a more general class of solutions forX =R ⁿ .Partially supported by M.U.R.S.T. Research funds (40%)Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

On the Radon-Nikodym property for operator valued measures

A theorem about the Radon-Nikodym property and the convergence of bounded martingales is proved for a bilinear integral in locally convex spaces.     

On the Friedrichs extension of some block operator matrices

We give a matrix representation for the resolvent of the Friedrichs extension of some semibounded 2×2 operator matrices and study their essential spectrum.     

On the peripheral spectrum of monic operator polynomials with positive coefficients

I wish to thank R. Nagel for his guidance and suggestions in the preparation of this paper. Also, I would like to thank G. Greiner and F. Räbiger for many interesting and helpful discussions.     

On orthogonally additive mappings,IV

Summary The conditional Cauchy functional equationF: (X, +, ) (Y, +), F(x + y) = F(x) + F(y) x, y X, x y, has first been studied under regularity (mainly continuity and boundedness) conditions and by referring to the inner product and the Birkhoff—James orthogonalities (A. Pinsker 1938, K. Sundaresan 1972, S. Gudder and D. Strawther 1975). The latter authors proposed an axiomatic framework for the space (X, +, ), and it then became possible to modify their axioms so that it could be proved without any regularity condition that the odd solutions of (*) are additive and the even ones are quadratic (cf., e.g., ([8], [12]). The results obtained included the classical case of the inner product orthogonality as well as the three following generalizations thereof: (i) Birkhoff—James orthogonality on a normed space, (ii) orthogonality induced by a non-isotropic sesquilinear functional, (iii) semi-inner product orthogonality.Making a further step in the modifications of the axioms for the space (X, +, ), the additive/quadratic representation of the solutions of (*) now can be proved in a much more general situation which includes also the case of the orthogonality induced by an isotropic symmetric bilinear functional.     

Exponential-cosine operator-valued functional equation in the strong operator topology

LetX be a Banach Space and letB(X) denote the family of bounded linear operators onX. LetR ⁺ = [0, ). A one parameter family of operators {S(t);t R ⁺},S:R ⁺ B(X), is called exponential-cosine operator function ifS(O) =I andS(s +t) – 2S(s)S(t) = (S(2s) – 2S ²(s))S(t –s), for alls, t R ⁺,s t. Let ,fD(A), and ,fD(B). It is shown that for a strongly continuous exponential-cosine operator {S(t)},fD(A ²) implies ₀ ^t (t –u(S(u)fduD(B) and B ₀ ^t (t –u)S(u)fdu =S(t)f –f +tAf – 2A ₀ ^t S(u)fdu + 2A ² ₀ ^t (t –u)S(u)fdu.D(B) is seen to be dense inD(A ²). Some regularity properties ofS(t) have also been obtained.