A Remark on Cosine Families

A Remark on Isotrivial Families

In the paper [1], the second named author proved that you can reduce a specific problem on finding rational points on Brauer-Severi varieties or more precisely on “families of Grassmanians” to the case where the discriminant locus is empty. This is explained in Section 7 of that paper, allthough what is written there is somewhat technical. The new result in this note is that this is actually a ...

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 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(...

A Remark on a Remark by Macaulay or Enhancing Lazard Structural Theorem

Zero-two law for cosine families

For (C(t))t≥0 being a strongly continuous cosine family on a Banach space, we show that the estimate lim supt→0+ ‖C(t) − I‖ < 2 implies that C(t) converges to I in the operator norm. This implication has become known as the zero-two law. We further prove that the stronger assumption of supt≥0 ‖C(t) − I‖ < 2 yields that C(t) = I for all t ≥ 0. For discrete cosine families, the assumption supn∈N ...