Compactness properties in the theory of the cosine operator functions (original) (raw)
On Certain Operator Families Related to Cosine Operator Functions
Taiwanese Journal of Mathematics, 1997
This paper is concerned with two cosine-function-related functions which are called cosine step response and cosine cumulative output. We study some of their properties, such as measurability, continuity, infinitesimal operator, compactness, positivity, almost periodicity, and asymptotic behavior.
On the convergence and approximation of cosine functions
Aequationes Mathematicae, 1974
201 1. Let X be a Banach space with norm II" It. M (X) denotes the space of all bounded linear operators on X. A cosine function is afamily of operators C=(C(t):teR= (-~, ~)) ~(X) satisfying (i) C(t+s)+C(t-s)=2C(t) C(s) for all t, seR, (ii) C (0) = I (= the identity operator), (iii) C(-)x: R ~ X is (strongly) continuous for each x~X. The (infinitesimal) generator A of a cosine function C is the operator A = C" (0). Here ' means differentiation with respect to t. The domain of A is D (A) = {xeX: the second strong derivative C"(t)x exists at t=0). Cosine functions are important since they bear the same relation to second order Cauchy problems as do semigroups to first order Cauchy problems. Specifically, the Cauchy problem for u" (t) = Au (t) (t ~ R) is well-posed if and only if A is the generator of a cosine function. (Cf. Fattorini [2, 3] for a precise formulation of this and for further results along this line.) The following generation theorem, due to Sova [9], DaPrato and Giusti [1], and Fattorini [2], is the cosine function analog of the Hille-Yosida semigroup generation theorem. THEOREM O. A is the generator of a cosine function C if and only if A is closed, densely defined, and there are constants M>0, to>~0 such that for 1>to, 22 is in the resolvent set of A and II (am/dim) [2 (22/-A)-1] II <~ Mm ! (2-co)-'~-i for all m~Z + = {0, 1, 2,...}. In this case, IfC(t)ll ~< Me ~'N for all teR, and cJO 2(221-a)-'x = f e-arC(t) xdt itl o for all 1>co and x~X.
On unbounded hyponormal operators II
Integral Equations and Operator Theory, 1992
Jal2 J alla~ The paper deals with unbounded hyponormal operators. Among others it is proved that any closed hyponormal operator with spectrum contained in a parabola generates a cosine function.
Cosine operator functions in R^2
Gulf Journal of Mathematics
In this paper, we consider the topic from the theory of cosine operator functions in 2-dimensional real vector space, which is an interplay between functional analysis and matrix theory. For the various cases of a given real matrix A= [α , β; γ , δ] we find out the appropriate cosine operator function C(t)= [a(t), b(t); c(t), d(t)], (t \in R) in a real vector space R2 as the solutions of the Cauchy problem C''(t)=AC(t), C(0)=I, C'(0)=0.
Proceedings of The American Mathematical Society, 1992
We study some problems related to convergence and divergence a.e. for Fourier series in systems {φ k }, where {φ k } is either a system of orthonormal polynomials with respect to a measure dµ on [−1, 1] or a Bessel system on [0, 1]. We obtain boundedness in weighted L p spaces for the maximal operators associated to Fourier-Jacobi and Fourier-Bessel series. On the other hand, we find general results about divergence a.e. of the Fourier series associated to Bessel systems and systems of orthonormal polynomials on [−1, 1].
A note on compact operators and operator matrices
1999
In this note two properties of compact operators acting on a separable Hilbert space are discussed. In the first part a characterization of compact operators is obtained for bounded operators represented as tri-block diagonal matrices with finite blocks. It is also proved that one can obtain such a tri-block diagonal matrix representation for each bounded operator starting from any orthonormal basis of the underlying Hilbert space by an arbitrary small Hilbert-Schmidt perturbation. The second part is devoted to the so-called Hummel's property of compact operators: each compact operator has a uniformly small orthonormal basis for the underlying Hilbert space. The class of all bounded operators satisfying Hummel's condition is determined.
A characterization of uniformly bounded cosine functions generators
Integral Equations and Operator Theory, 1989
The infinitesimal generator A of C is the closed and densely defined ope-"x with its natural domain. rator given by Ax: = C o C is called uniformly bounded if there is a constant M > 1 withll CtlI<M,t~R. Uniformly bounded cosine functions in Hilbert spaces were studied by H.O. Fattorini who proved that * Suported by DAAD, West Germany.
On the Operator-valued mu\mumu-cosine functions
arXiv (Cornell University), 2017
Let (G, +) be a topological abelian group with a neutral element e and let µ : G -→ C be a continuous character of G. Let (H, •, • ) be a complex Hilbert space and let B(H) be the algebra of all linear continuous operators of H into itself. A continuous mapping Φ : G -→ B(H) will be called an operator-valued µ-cosine function if it satisfies both the µ-cosine equation Φ(x + y) + µ(y)Φ(xy) = 2Φ(x)Φ(y), x, y ∈ G and the condition Φ(e) = I, where I is the identity of B(H). We show that any hermitian operator-valued µ-cosine functions has the form 2 where Γ : G -→ B(H) is a continuous multiplicative operator. As an application, positive definite kernel theory and W. Chojnacki's results on the uniformly bounded normal cosine operator are used to give explicit formula of solution of the cosine equation.
Which operators approximately annihilate orthonormal basesΦ
Acta Scientiarum Mathematicarum
A theorem of P. A. Fillmore and J. P. Williams [Adv. Math. 7, 254-281 (1971; Zbl 0224.47009)] implies that a bounded operator A on a separable Hilbert space H is compact if and only if it satisfies lim n Ae n =0, or equivalently, lim n (Ae n |e n )=0 for each orthonormal basis (e n ) for H. In the present note this theorem is reproved using the fact observed by Halmos that each sequence of unit vectors weakly converging to 0 approximately contains an orthonormal subsequence. It is also noted that the stronger version of the theorem remains true, namely without the continuity assumption on A. In the second part of the note the bounded operators for which there exists an orthonormal basis such that either of the above equalities holds are exhibited and completely described.