Bounded Cosine Functions Close to Continuous Scalar Bounded Cosine Functions

Abstract

Let (C (t )) t ∈R be a cosine function in a unital Banach algebra. We show that if sup t ∈R C (t ) − c(t ) < 2 for some continuous scalar bounded cosine function (c(t )) t ∈R , then the closed subalgebra generated by (C (t )) t ∈R is isomorphic to C k for some positive integer k. If, further, sup t ∈R C (t )−c(t ) < 8 3 3, then C (t ) = c(t ) for t ∈ R.

A Zero- Law for Cosine Families

“…In fact for every a ∈ R there exists a largest constant k(a) such that sup n≥1 |cos(nb)− cos(na)| < k(a) implies that cos(nb) = cos(na) for n ≥ 1, and we prove that if a cosine sequence (C (n)) n∈Z in a Banach algebra A satisfies sup n≥1 |C (n) − cos(na)1 A | < k(a) then C (n) = cos(na) for n ≥ 1. This follows from the following result, proved by the author in [10]. Theorem 1.1.…”

“…The proofs were based on rather involved arguments from operator theory and semigroup theory. Very recently, Bobrowski, Chojnacki and Gregosiewicz [5] showed more precisely that if a cosine function C = C (t ) satisfies sup t ∈R C (t )−c(t ) < 8 3 3 for some scalar bounded continuous cosine function c(t ), then C (t ) = c(t ) for t ∈ R, without any continuity assumption on C , and the same result was obtained independently by the author in [10]. The constant 8 3 3 is obviously optimal, since sup t ∈R |cos(at ) − cos(3at )| = 8 3 3 for every a ∈ R \ {0}.…”

“…We discuss now the case where a π and b π are both rational, with b ∉ ±a + 2πZ. if q = 3p, see for example [10]. Now let µ < λ, and let η < δ be two real numbers such that (2)…”

A short proof of the zero-two law for cosine functions

“…The zero-two law for scalar cosine functions pertains to folklore, but we could not find a reference in the litterature for the following certainly well-known lemma, which is a variant of proposition 3.1 of [7].…”

“…This question was answered positively by Chojnacki in [5]. Using a sophisticated argument based on ultrapowers, Chonajcki deduced this zero-two law from the fact that if a cosine sequence C (t ) satisfies sup t ∈R C (t ) − 1 A < 2, then C (t ) = 1 A for t ∈ R. This second result, which was obtained independently by the author in [7], is proved by Chojnacki in [5] by adapting methods used by Bobrowski, Chojnacki and Gregosiewicz in [3] to show that if a cosine sequence (C (t )) t ∈R satisfies sup t ∈R C (t ) − cos(at )1 A < 8 3 3 for some a ∈ R, then C (t ) = cos(at )1 A for t ∈ R, a result also obtained independently by the author in [7], which improves previous results of [2], [4] and [10].…”