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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 \(E\).

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