A Note on Positivity of One-Dimensional Elliptic Differential Operators

Allaberen Ashyralyev, Sema Akturk

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.

Keywords

Positive operator; fractional spaces; Green's function, H\"{o}lder spaces

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