On the domain of some ordinary differential operators in spaces of continuous functions (Q1881507)

The following second-order ordinary differential operator \(Au=au''+ bu'\) with unbounded coefficients in spaces of continuous functions is studied. A description of the domain on which \(A\) generates a semigroup is given. The authors deal with \(C_b(\mathbb R)\) (the space of bounded and continuous functions in \(\mathbb R)\) and with \(C(\overline{\mathbb R})\), (the space of continuous functions having finite limits at \(\pm\infty)\). The operator \(A\) is considered with its maximal domain in \(C_b(\mathbb R)\): \[ D_{\max} (A):= \bigl\{u\in C_b(\mathbb R)\cap C^2(\mathbb R): Au \in C_b(\mathbb R)\bigr\} \] and throughout the paper the following condition: \((H_0)\) \(\lambda-A\) is injective on \(D_{\max}(A)\) for some \(\lambda>0\), is assumed. This is equivalent to saying that \((A,D_{\max}(A))\) generates a semigroup of positive contractions in \(C_b(\mathbb R)\), which is not however strongly continuous. The condition \((H_0)\) implies that \(\lambda-A\) is injective on \(D_{\max}(A)\) for all \(\lambda>0\). Moreover it turns out that \(\lambda-A\) is injective on \(D_{\max}(A)\) if and only if it is injective on \(D_m(A)\), where \[ D_m(A)=\bigl\{u\in C(\overline{\mathbb R})\cap C^2 (\mathbb R):Au\in C(\overline{\mathbb R}) \bigr\} \] is the maximal domain in \(C(\overline{\mathbb R})\). It follows that \((A,D_m (A))\) generates a strongly continuous semigroup of positive contractions in \(C(\overline{\mathbb R})\). Under suitable conditions, the main results show that \[ D_{\max}(A)= \bigl\{u\in C^2_b({\mathbb R}): au'',bu'\in C_b(\mathbb R)\bigr\} \] and, if \(a\) is bounded, then \[ D_m(A)=\bigl\{u\in C^2(\overline{\mathbb R}):bu'\in C (\overline {\mathbb R})\bigr\}. \]