Weak conditions for generation of cosine families in linear topological spaces

Let $A$ be a closed linear operator in a Banach space $X$. Weak conditions are found under which (I) the abstract Cauchy problem in $X$:

$$u''\left( t \right) = Au\left( t \right),\quad t \in R;\quad u\left( 0 \right) = {u_0},\quad u'\left( 0 \right) = {u_1}$$

has a unique solution for each ${u_0}$ and ${u_1}$ given in a dense subset $Y$ of $X$, and (II) the set $Y$ becomes a linear topological space where $A{\vert _Y}$ generates a continuous cosine family.

The conditions are satisfied for example by the generator of a strongly continuous or holomorphic semigroup in $X$.