The structure of fractional spaces generated by a two-dimensional neutron transport operator and its applications (Q1790583)

In this paper, a densely defined operator $B$ in a Banach space $E$ is called \textit{positive} if its spectrum lies in a sector $|\arg\lambda| \le \varphi,$ $0 \le \varphi < \pi$ and \[ \| (\lambda I - B)^{-1} \| \le \frac{M}{1 + |\lambda|}, \quad | \arg \lambda| > \varphi \, . \] The operator to which this definition is applied is $B = -A,$ where $A$ is the neutron transport operator \[ Au(x, y, \omega_1, \omega_2) = \omega_1 \frac{\partial u}{\partial x}+ \omega_2 \frac{\partial u}{\partial y} \] in $\mathbb{R}^2,$ with $\omega_1, \omega_2$ direction cosines. The first result is positivity of $B$ in the spaces $C(\mathbb{R}^2)$ and $L^p(\mathbb{R}^2),$ $1 \le p < \infty.$ Then, the space $E_\alpha(C(\mathbb{R}^2), B)$ is introduced in terms of the growth of $\|B(\lambda I + B)f\|_{C({\mathbb{R}^2})}$ as $\lambda \to \infty$ and $B$ is proved positive in $E_\alpha(C(\mathbb{R}^2), B),$ hence in a Hölder space $C^\alpha(\mathbb{R})$ with equivalent norm. Finally, the same results are shown for the space $E_\alpha^p (L^p(\mathbb{R}^2), B)$ defined as $E_\alpha(C(\mathbb{R}^2), B)$ but replacing $C(\mathbb{R}^2)$ by $L^p(\mathbb{R}^2),$ and for a Sobolev-Slobodecki space $W^p_\alpha(\mathbb{R}^2)$ of equivalent norm.