A characteristic of cosine operat or functions in Banach spaces

In the present pater,some relationship between cosine operator functions in Banach spaces and the Cauchy problem of second order undamped differential equations was studied,and by means of the existence and uniqueness of the Cauchy problem,acharacterization of the infinitesimal generator of cosine operator functions which is different from the one of Hille-Yosida type was proposed.