On the theory of remediability
Volume 63 / 2003
Abstract
Suppose $\{G_1(t)\}_{t\ge 0}$ and $\{G_2(t)\}_{t\ge 0}$ are two families of semigroups on a Banach space $X$ (not necessarily of class $C_0$) such that for some initial datum $u_0$, $G_1(t)u_0$ tends towards an undesirable state $u^*$. After remedying by means of an operator $\rho$ we continue the evolution of the state by applying $G_2(t)$ and after time $2t$ we retrieve a prosperous state $u$ given by $u=G_2(t) \rho G_1(t) u_0.$ Here we are concerned with various properties of the semigroup ${\cal G}(t):\;\rho \rightarrow G_2(t)\rho G_1(t).$ We define ${\cal R}(X)$ to be the space of remedial operators for $G_1(t)$ and $G_2(t)$, when the above map is well defined for all $\rho \in {\cal R}(X)$ and satisfies the properties of a uniformly bounded semigroup on ${\cal R}(X)$. In this paper we study some properties of the space ${\cal R}(X)$ and we prove that when $A_i$ generate a regularized semigroup for $i=1,2$, then the operator $\Delta$ defined on ${\cal L}(X)$ by $\Delta \rho= A_2\rho + \rho A_1$ generates a tensor product regularized semigroup. Finally, we give two examples of remedial operators in radiotherapy and chemotherapy in proliferation of cancer cells.
