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 \\ $D(A)=\{u:u_{tt},u\in C\left( \mathbb{R}_{+}\right) ,u(0)=0,u(\infty )=0\},$ where $\mathbb{R}_{+}=\left[ 0,\infty \right) .$ 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.