##
**The structure of fractional spaces generated by a two-dimensional neutron transport operator and its applications.**
*(English)*
Zbl 1435.47045

In this paper, a densely defined operator \(B\) in a Banach space \(E\) is called 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.

Reviewer: Hector O. Fattorini (Los Angeles)

### MSC:

47B65 | Positive linear operators and order-bounded operators |

47F05 | General theory of partial differential operators |

47B38 | Linear operators on function spaces (general) |

47B01 | Operators on Banach spaces |

35A35 | Theoretical approximation in context of PDEs |

35K30 | Initial value problems for higher-order parabolic equations |

34B27 | Green’s functions for ordinary differential equations |

35Q49 | Transport equations |

46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |

### Keywords:

neutron transport operator; fractional space; Slobodeckij space; positive operator; Hölder spaces; Sobolev-Slobodeckiĭ spaces; positive operators
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\textit{A. Ashyralyev} and \textit{A. Taskin}, Adv. Oper. Theory 4, No. 1, 140--155 (2019; Zbl 1435.47045)

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