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In this study, the second order differential operator \(A^{x}\) defined by the formula \(A^{x}u=-u_{xx}(x)+\delta u(x)\), \(\delta \geq 0\), with domain \(D(A^{x})= \{ u(x):u(x),u^{\prime }(x),u^{\prime \prime } (x)\in C(R^{1}), u(x)=u(x+2\pi ),x\in R^{1}, \int_{0}^{2\pi}u(x)\,dx=0 \} \) is considered. The Green function of the differential operator \(A^{x}\) is constructed. The estimates for the Green function are obtained. It is proved that for any \(\alpha\in(0,\frac{1}{2})\), the norms in the spaces \(E_{\alpha }=E_{\alpha}(\mathring{C} (R^{1} ),A^{x})\) and \(\mathring{C}^{2\alpha} (R^{1} )\) are equivalent.

A note on fractional spaces generated by the positive operator with periodic conditions and applications