Quasi-compactness and absolutely continuous kernels

We show how the essential spectral radius r _e (Q) of a bounded positive kernel Q, acting on bounded functions, is linked to the lower approximation of Q by certain absolutely continuous kernels. The standart Doeblin’s condition can be interpreted in this context, and, when suitably reformulated, it leads to a formula for r _e (Q). This results may be used to characterize the Markov kernels having a quasi-compact action on a space of measurable functions bounded with respect to some test function, when no irreducibilty and aperiodicity are assumed.      

Semigroups for flows in infinite networks

Inspired by previous work of M. Kramar and E. Sikolya (Math. Z. 249, 139–162, [2005]), we study transport processes on infinite networks. These “flows” can be modeled by operator semigroups on a suitable Banach space. Using functional analytical and graph theoretical methods, we investigate its spectral properties to determine the long time behavior of the system, and finally characterize uniform convergence of the semigroup to a periodic group under appropriate assumptions on the network.