On contractions in \(L_ 1\). (English) Zbl 0702.47002

Mesiar, Radko

Čas. Pěstování Mat. 114, No. 4, 337-342 (1989).

Summary: Let (X,\({\mathcal S},m)\) be a \(\sigma\)-finite measure space, T a contraction on \(L_ 1(X)\), \(f\in L_ 1(X)\). For a given nondecreasing sequence \(\{a_ n\}\) of positive reals we study the pointwise convergence of \(T^ nf/a_ n\). If a series \(\sum 1/a_ n\) is convergent, then \(T^ nf/a_ n\to 0\), a.e. For a divergent series \(\sum 1/a_ n\) we establish a condition which enables us to construct a contraction P on \(L_ 1(<0,1))\) and \(f\in L_ 1(<0,1))\) such that lim sup \(P^ nf/a_ n=\infty\), a.e.

MSC:

47A35

Ergodic theory of linear operators

Keywords:

pointwise convergence; \(\sigma \) -finite measure space; contraction