On spectral gaps of growth-fragmentation semigroups in higher moment spaces

Abstract

<p style='text-indent:20px;'>We present a general approach to proving the existence of spectral gaps and asynchronous exponential growth for growth-fragmentation semigroups in moment spaces <inline-formula><tex-math id="M1">\begin{document}$ L^{1}( \mathbb{R} _{+};\ x^{\alpha }dx) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ L^{1}( \mathbb{R} _{+};\ \left( 1+x\right) ^{\alpha }dx) $\end{document}</tex-math></inline-formula> for unbounded total fragmentation rates and continuous growth rates <inline-formula><tex-math id="M3">\begin{document}$ r(.) $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M4">\begin{document}$ \int_{0}^{+\infty } \frac{1}{r(\tau )}d\tau = +\infty .\ $\end{document}</tex-math></inline-formula> The analysis is based on weak compactness tools and Frobenius theory of positive operators and holds provided that <inline-formula><tex-math id="M5">\begin{document}$ \alpha &gt;\widehat{\alpha } $\end{document}</tex-math></inline-formula> for a suitable threshold <inline-formula><tex-math id="M6">\begin{document}$ \widehat{ \alpha }\geq 1 $\end{document}</tex-math></inline-formula> that depends on the moment space we consider. A systematic functional analytic construction is provided. Various examples of fragmentation kernels illustrating the theory are given and an open problem is mentioned.</p>