Solvability of an operator Riccati integral equation in a reflexive Banach space (Q2319429)

The main result of the paper is that if \(X\) is a reflexive Banach space then there is a unique, strongly continuous solution \(P\) to the Riccati integral equation \[ P(t)=\overrightarrow{U}_{t,T}G\overleftarrow{U}_{T,t}+\int_t^T \overrightarrow{U}_{t,s}\{C(s)-P(s)B(s)P(s)\}\overleftarrow{U}_{s,t}\, ds. \] Here \(P(t)\) belongs to the space \(\mathcal L(X,X^*)\) of bounded linear operators from \(X\) to its dual \(X^*\) and is self-adjoint and nonnegative, \(\{\overrightarrow{U}_{t,s}\}\) and \(\{\overleftarrow{U}_{t,s}\} = \{\overrightarrow{U}_{s,t}^*\}\) are strongly continuous uniformly bounded forward and backward evolution families on \(\mathcal L(X)\) and \(\mathcal L(X^*)\), respectively, the operator functions \(C\) and \(B\) with values in \(\mathcal L(X,X^*)\) and \(\mathcal L(X^*,X)\), respectively, are strongly continuous, self-adjoint and nonnegative, and \(G\) belongs to \(\mathcal L(X,X^*)\). In contrast to earlier papers, no assumption about an embedding between the space \(X\) and its dual is made.