Cosine operator functions in R^2 (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 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.

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

Journal of Soviet Mathematics, 1991

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. differenHable 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 perturbalion.~ for OS's and COF'.s, adjoint OS's and COF's.

Forms, functional calculus, cosine functions and perturbation

Perspectives in Operator Theory, 2007

In this article we describe properties of unbounded operators related to evolutionary problems. It is a survey article which also contains several new results. For instance we give a characterization of cosine functions in terms of mild well-posedness of the Cauchy problem of order 2, and we show that the property of having a bounded H ∞-calculus is stable under rank-1 perturbations whereas the property of being associated with a closed form and the property of generating a cosine function are not.

Convoluted C-operator families and abstract Cauchy problems

We present the basic structural properties of convoluted C-cosine functions and semigroups in a Banach space setting and consider the corresponding abstract Cauchy problems. Notation. By E and L(E) are denoted a complex Banach space and Banach algebra of bounded linear operators on E. For a closed linear operator A on E, D(A), Kern(A), R(A), ρ(A) denote its domain, kernel, range and resolvent set, respectively. Put D ∞ (A) := n∈N 0 D(A n). By [D(A)] is denoted the Banach space D(A) endowed with the graph norm. In this paper, C ∈ L(E) is an injective operator satisfying CA ⊂ AC. The C-resolvent set of A, denoted by ρ C (A), is defined by ρ C (A) := {λ ∈ C : R(C) ⊂ R(λ − A) and λ − A is injective}. Further on, in some statements which are to follow we use that a complex valued function K ∈ L 1 loc ([0, τ)), 0 < τ ≤ ∞ is a kernel which means that for every φ ∈ C([0, τ)), the assumption t 0

Cosine of angle and center of mass of an operator

Mathematica Slovaca, 2012

We consider the notion of real center of mass and total center of mass of a bounded linear operator relative to another bounded linear operator and explore their relation with cosine and total cosine of a bounded linear operator acting on a complex Hilbert space. We give another proof of the Min-max equality and then generalize it using the notion of orthogonality of bounded linear operators. We also illustrate with examples an alternative method of calculating the antieigenvalues and total antieigenvalues for finite dimensional operators.