Representation of positive operators and alternating sequences (Q1175767)

It is proved that if \(T\) is a positive contraction on \(L^ p\), then there exist a positive contraction \(U\) on \(L^ 1\) and a positive contraction \(V\) on \(L^ \infty\) such that for any non-negative functions \(\alpha\) and \(\beta\) \(T(\alpha\beta)\leq(U\alpha^ p)^{1/p}(V\beta^ q)^{1/q}\). In this case \((U,V)\) is called a representing pair for \(T\). A contraction \(S\) on \(L^ 2\) is called a weak adjoint of \(T\) if \(S\) and \(T^*\) have a common representing pair. The main result of the paper under review claims that if \(\{T_ k\}\) is a sequence of positive contractions on \(L^ p\) and \(\{S_ n\}\) is a sequence of positive contractions on \(L^ r\) such that \(S_ n\) is a weak adjoint of \(T_ n\) for each \(n\), then for any non-negative \(f\) in \(L^ p\) the sequence \[ S_ 1\dots S_ n(T_ 1\dots T_ n f)^{p/2} \] converges a.e.