On a characteristic of cosine functions

This paper is concerned with the property of cosine function. It is proved that a family {T(t)}(t >= 0) of strongly continuous linear operators is a cosine function on Banach space X if and only if T(0)=I and there holds integral(t+s)(0) T(tau)d tau = T(t) integral(s)(0) T(tau)d tau + integral(t)(0) T(tau)d tau T(s), t, s >= 0, where all the integrals concerning operator valued functions are understood to be in the strong operator topology.