# zbMATH — the first resource for mathematics

On the solution of a problem involving the abstract cosine function. (English. Russian original) Zbl 0879.47024
Russ. Acad. Sci., Dokl., Math. 49, No. 3, 556-559 (1994); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 336, No. 5, 584-586 (1994).
Let $$E$$ be a Banach space. Definition. A family of bounded linear operators $$C(t)$$ in $$E$$ is called a strongly continuous cosine function (COF) if it satisfies the following conditions
1) $$C(t+s) +C(t-s) =2C(t) C(s)$$, $$t,s\in R$$;
2) $$C(0) =I$$;
3) $$C(t)x$$ is strongly continuous with respect to $$t$$ for any $$x\in E$$.
For a COF the generating operator $$A$$ is defined by $Ax= \lim_{h\to 0} {1\over 2h^2} \bigl[C(h) -I\bigr] x\quad \text{for } x\in E.$ The author proves that if $$A$$ is a generator of a COF in $$E$$, then every operator polynomial $$P_{2m+1} (A)= A^{2m+1} +\sum^{2m}_{k=0} c_kA^k$$, $$c_k\in R$$, $$m=0,1, \dots$$, is also a generator of a COF in $$E$$.

##### MSC:
 47D09 Operator sine and cosine functions and higher-order Cauchy problems