Cosine families and semigroups really differ

Cosine Families and Semigroups Really Dif- fer

We reveal three surprising properties of cosine families, distinguishing them from semigroups of operators: (1) A single trajectory of a cosine family is either strongly continuous or not measurable. (2) Pointwise convergence of a sequence of equibounded cosine families implies that the convergence is almost uniform for time in the entire real line; in particular, cosine families cannot be pert...

On Cosine Families Close to Scalar Cosine Families

A classic result, in its early form due to Cox [2], states that if A is a normed algebra with a unity denoted by 1 and a is an element of A such that supn∈N ‖a− 1‖ < 1, then a = 1. Cox’s version concerned the case of square matrices of a given size. This was later extended to bounded operators on Hilbert space by Nakamura and Yoshida [6], and to an arbitrary normed algebra by Hirschfeld [4] and...

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 ...

On Multivalued Cosine Families

Let K be a convex cone in a real Banach space. The main purpose of this paper is to show that for a regular cosine family {Ft : t ∈ R} of linear continuous multifunctions Ft : K → cc(X) there exists a linear continuous multifunction H : K → cc(K) such that Ft(x) ⊂ ∞ X n=0 t (2n)! H(x). Let X be a real normed vector space. We will denote by n(X) the family of all nonempty subsets of X and by cc(...

Convoluted C-cosine Functions and Convoluted C-semigroups