p ON SPECTRAL GAPS OF GROWTH-FRAGMENTATION SEMIGROUPS IN HIGHER MOMENT SPACES

We present a general approach to proving the existence of spectral gaps and asynchronous exponential growth for growth-fragmentation semigroups in moment spaces L1 (R+; x alpha dx) and L1(R+; (1 + x)alpha dx) for unbounded total fragmentation rates and continuous growth rates r(.) such that f +infinity r(Tau) d Tau = +infinity. The analysis is based on weak compactness tools and Frobenius theory of positive operators and holds provided that alpha > alpha b for a suitable threshold alpha b >= 1 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.