Extended groups of semigroups and backward problems of heat equations

In this paper, we are concerned with backward solvabilities of heat equations, in an abstract framework. We show that semigroups $T_t$ in Banach spaces $X$, generated by heat operators, are extendable to groups in an extended space $E$, which is obtained by considering a sequence of wider Banach spaces containing $X$, i.e. $X$$/subset$$X_t$$/subset$$X_s$... $(t<s)$, under the following two conditions. One is the density assumption on a subset $D$ of $X$, the set of initial values $x$ from which $T_{-t}x$ exists for all $t>0$. Another is the backward uniqueness of the semigroup $T_t$. For example, we prove the holomorphic semigroup satisfies the above conditions, and thus is extendable to a group in a larger functional space $E$. We also studied structual properties of the extended space $E$.