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Published online by Cambridge University Press:
**17 April 2017**

Let $a\in \mathbb{R}$ , and let $k(a)$ be the largest constant such that $\sup |\text{cos}(na)-\cos (nb)|<k(a)$ for $b\in \mathbb{R}$ implies that $b\in \pm a+2\unicode[STIX]{x1D70B}\mathbb{Z}$ . We show that if a cosine sequence $(C(n))_{n\in \mathbb{Z}}$ with values in a Banach algebra $A$ satisfies $\sup _{n\geq 1}\Vert C(n)-\cos (na).1_{A}\Vert <k(a)$ , then $C(n)=\cos (na).1_{A}$ for $n\in \mathbb{Z}$ . Since $\!\sqrt{5}/2\leq k(a)\leq 8/3\!\sqrt{3}$ for every $a\in \mathbb{R}$ , this shows that if some cosine family $(C(g))_{g\in G}$ over an abelian group $G$ in a Banach algebra satisfies $\sup _{g\in G}\Vert C(g)-c(g)\Vert <\!\sqrt{5}/2$ for some scalar cosine family $(c(g))_{g\in G}$ , then $C(g)=c(g)$ for $g\in G$ , and the constant $\!\sqrt{5}/2$ is optimal. We also describe the set of all real numbers $a\in [0,\unicode[STIX]{x1D70B}]$ satisfying $k(a)\leq \frac{3}{2}$ .

cosine sequence
scalar cosine sequence
Kronecker’s theorem
commutative Banach algebra
cyclotomic polynomial

Primary:
46J45: Radical Banach algebras

Type

Research Article

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© 2017 Australian Mathematical Publishing Association Inc.

Arendt, W., Batty, C. J. K., Hieber, M. and Neubrander, F., Vector-Valued Laplace Transforms and Cauchy Problems (Birkhäuser, Basel, 2001).CrossRefGoogle Scholar

Arkhangel’skii, A. V. and Ponomarev, V. I., Fundamentals of General Topology: Problems and Exercises (D. Reidel Publishing Company/Hindustan Publishing Company, 1984), (translated from Russian).Google Scholar

Batkai, A., Engel, K.-J. and Haase, M., ‘Cosine families generated by second order differential operators on *W*
^{1, 1}(0, 1) with generalized Wentzell boundary conditions’, Appl. Anal.
84 (2005), 867–876.CrossRefGoogle Scholar

Bobrowski, A. and Chojnacki, W., ‘Isolated points of some sets of bounded cosine families, bounded semigroups, and bounded groups on a Banach space’, Studia Math.
217 (2013), 219–241.CrossRefGoogle Scholar

Bobrowski, A., Chojnacki, W. and Gregosiewicz, A., ‘On close-to-scalar one-parameter cosine families’, J. Math. Anal. Appl.
429 (2015), 383–394.CrossRefGoogle Scholar

Chojnacki, W., ‘On cosine families close to scalar cosine families’, J. Aust. Math. Soc.
99 (2015), 166–174.CrossRefGoogle Scholar

Cox, R. H., ‘Matrices all of whose powers lie close to the identity’, Amer. Math. Monthly
73 (1966), 813.Google Scholar

Dales, H. G., Banach Algebras and Automatic Continuity, London Mathematical Society Monographs, 24 (Clarendon Press, Oxford, 2001).Google Scholar

Esterle, J., ‘Bounded cosine functions close to continuous scalar bounded cosine functions’, Int. Eq. Op. Th.
85 (2016), 347–357.CrossRefGoogle Scholar

Haase, M., ‘The group reduction for bounded cosine functions on UMD spaces’, Math. Z.
262(2) (2009), 281–299.CrossRefGoogle Scholar

Haase, M., ‘The functional calculus approach to cosine operator functions’, in: Recent Trends in Analysis, Proceedings of the Conference in honour of N. K. Nikolski held in Bordeaux 2011 (Theta Foundation, Bucharest, 2013), 123–147.Google Scholar

Hirschfeld, R. A., ‘On semi-groups in Banach algebras close to the identity’, Proc. Japan Acad. Ser. A Math. Sci.
44 (1968), 755.Google Scholar

Kahane, J. P. and Salem, R., Ensembles Parfaits et Séries Trigonométriques (Hermann, Paris, 1963).Google Scholar

Nagy, B., ‘Cosine operator functions and the abstract Cauchy problem’, Period. Math. Hungar.
7 (1976), 15–18.CrossRefGoogle Scholar

Nakamura, M. and Yoshida, M., ‘On a generalization of a theorem of Cox’, Proc. Japan Acad. Ser. A Math. Sci.
43 (1967), 108–110.Google Scholar

Schwenninger, F. and Zwart, H., ‘Less than one, implies zero’, Studia Math.
229 (2015), 181–188.Google Scholar

Schwenninger, F. and Zwart, H., ‘Zero-two law for cosine families’, J. Evol. Equ.
15 (2015), 559–569.CrossRefGoogle Scholar

Travis, C. and Webb, G., ‘Cosine families and abstract nonlinear second order differential equation’, Acta Math. Acad. Sci. Hungar.
32 (1978), 75–96.CrossRefGoogle Scholar

Wallen, L. J., ‘On the magnitude of ∥*x*
^{
n
} - 1∥ in a normed algebra’, Proc. Amer. Math. Soc.
18 (1967), 956.Google Scholar

Washington, L. C., Cyclotomic Fields, 2nd edn, Graduate Texts in Mathematics, 83 (Springer, New York–Berlin–Heidelberg, 1997).CrossRefGoogle Scholar

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LAW FOR COSINE FAMILIES

- Volume 104, Issue 2

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$\!\sqrt{5}/2$
LAW FOR COSINE FAMILIES

- Volume 104, Issue 2

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