The Perron-Frobenius theory for almost periodic representations of semigroups in the spaces Lp
The classical Perron-Frobenius theory of nonnegative matrices is generalized to nonnegative almost periodic representations of topological semigroups in the spaces Lp(Ω, σ, Μ), where (Ω, σ, Μ) is a space with a σ-finite measure, 1≤p<∞. With each such representation one connects the associated action of its Sushkevich kernel onto some naturally arising space with measure; this allows that the investigation of the spectral properties of the representation be reduced to the investigation of the ergodic properties of the corresponding action. In particular, it is established, that the boundary spectrum of an indecomposable representation is a subgroup of the dual group of the Sushkevich kernel (coincides with it if the considered semigroup is Abelian). In the general case the boundary spectrum is cyclic (i.e., the union of the subgroups of the dual group of the Sushkevich kernel). The results of the paper are new even at the consideration of the semigroups of degree one of an operator (if other words, the representations of the semigroup Z+); this yields the generalized Perron-Frobenius theory for nonnegative a. p. operators.
KeywordsSpectral Property Dual Group Ergodic Property Topological Semigroup Nonnegative Matrice
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