An ergodic theorem on Banach lattices
A well known theorem of \textit{U. Krengel} [Z. Wahrscheinlichkeitstheorie Verw. Gebiete 6, 302-316 (1966; Zbl 0218.60060)] states that for any positive linear contraction T in \(L_ 1\) and for each \(f\in L_ 1\) the averages \(A_ nf=n^{-1}\sum^{n-1}_{k=0}T^ kf\) are convergent in probability. In the present paper the above theorem is extended to positive linear contractions T acting in a Banach lattice E. Instead of the convergence in probability, the authors propose to consider the existence of the truncated limit \(TL(A_ nf)=\Phi\), which means that \(\lim_{n} (A_ nf\wedge ku)=\Phi_ k\) exists in norm for some weak unit u and for each k, and that \(\Phi_ k\uparrow \Phi\). If \(E=L_ 1\) then this notion is equivalent to the convergence in probability. It is shown that \(TL(A_ nf)\) exists for each \(f\in E_+\) if E has a weak unit, is weakly sequentially complete, and satisfies an additional condition (C). In case of Orlicz spaces on a finite non-atomic measure space, the condition (C) is equivalent to the condition \(\Delta_ 2\).
scientific article; zbMATH DE number 3951341
scientific article