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On the second order Cauchy problem associated with a linear operator. (English) Zbl 0746.47023
Let $A$ be a linear operator in a Banach space $X$ whose resolvent set contains $(0,\infty)$. The author defines a subspace $W$ of initial values $x\in X$ for which the second order abstract Cauchy problem $$u''(t)=Au(t), \qquad u(0)=x, \qquad u'(0)=0$$ has a solution on $\bbfR$. The space $W$ is contained in the Hille-Yosida subspace introduced by {\it S. Kantorovitz} [Math. Ann. 282, 535-544 (1988)].

##### MSC:
 47D09 Operator sine and cosine functions and higher-order Cauchy problems
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##### References:
 [1] Da Prato, G.; Giusti, E.: Una caracterizzatione dei generatori di funzioni coseno astratto. Boll. uni. Mat. ital. 22, 357-362 (1967) · Zbl 0186.47702 [2] Fattorini, H. O.: Ordinary differential equations in linear topological spaces, I. J. differential equations 5, 72-105 (1968) · Zbl 0175.15101 [3] Fattorini, H. O.: Ordinary differential equations in linear topological spaces, II. J. differential equations 6, 50-70 (1969) · Zbl 0181.42801 [4] Hille, E.; Phillips, R. S.: Functional analysis and semi-groups. Amer. math. Soc. colloquium publ. 31 (1957) [5] Kantorovitz, S.: The hille-yosida space of an arbitrary operator. J. math. Anal. appl. 136, 107-111 (1988) · Zbl 0675.47033 [6] Kantorovitz, S.: Spectral representations for unbounded operators with real spectrum. Math. ann. 282, 535-544 (1988) · Zbl 0632.47030 [7] Sova, M.: Cosine operator functions. Rozprawy mat. 49, 1-46 (1966)