Ergodic theorem for positive operators on \(L_p\)-spaces \((1<p<\infty)\) revisited
Let \((\Omega, \Sigma, \mu)\) be a probability space and let \(1<p<\infty.\) A positive linear operator \(A\) on \(L^p(\Omega,\Sigma, \mu)\) is said to be subadditive with respect to powers (s.r.p.) if \(A^{m+n} <A^m +A^n\) for all \((m,n)\in {\mathbb N}^2\), where \(A^0=I.\) The author proves the following structure theorem for these operators. Let \(A\) be a positive linear s.r.p. operator on \(L^p(\Omega, \Sigma, \mu)\), then there exists a partition \(\Omega_1,\Omega_2,\Omega_3\) of \(\Omega\) into disjoint subsets such that \(A\) can be written as \[ A=(E_1+E_2+E_1AE_2)+(E_3AE_3), \] where \(E_if=f\cdot I_{\Omega_i}.\) As an application of this structure theorem, the author proves the following result. Let \(A\) be a positive linear s.r.p. operator on \(L^p(\Omega, \Sigma, \mu)\). If \(A^n/n\) converges to \(0\), then \(A^n\) is bounded for all \(n\), and for all \(f\in L^p(\Omega, \Sigma, \mu)\) the sequence \(A^nf\) converges almost everywhere.
scientific article; zbMATH DE number 1744379
scientific article