A Note on Positivity of One-Dimensional Elliptic Differential Operators

Abstract

We consider a structure of fractional spaces $E_{\alpha }(C\left({\mathbb{R}}_{+}\right),A)$ generated by the positive differential operator $A$ definedby the formula $Au(t)=-u_{tt}(t)+u(t)$ with domain \ $D(A)=\{u:u_{tt},u\in C\left({\mathbb{R}}_{+}\right),u(0)=0,u(\mathrm{\infty })=0\},$ where ${\mathbb{R}}_{+}=[0,\mathrm{\infty }).$ It is established that for any $0<\alpha <1/2,$the norms in the spaces $E_{\alpha }(C\left({\mathbb{R}}_{+}\right),A)$ and $C^{2\alpha }\left({\mathbb{R}}_{+}\right)$ are equivalent. The positivity of the differential operator $A$ in $C^{2\alpha }\left({\mathbb{R}}_{+}\right)$is established.