On cosine families close to scalar cosine families

TL;DR

This paper proves that two cosine families in a normed algebra, where one is scalar and bounded, must be identical if they are uniformly close in norm, highlighting a stability property.

Contribution

It establishes a uniqueness result for cosine families close to scalar families in normed algebras, extending understanding of their stability.

Findings

If two cosine families differ by less than 1 in norm, they are identical.

The result applies to families indexed by an Abelian group.

Scalar bounded cosine families are uniquely determined by their proximity.

Abstract

We prove that if two normed-algebra-valued cosine families indexed by a single Abelian group, of which one is bounded and comprised solely of scalar elements of the underlying algebra, differ in norm by less than $1$ uniformly in the parametrising index, then these families coincide.