On the entangled ergodic theorem

We study the convergence of the so-called entangled ergodic averages
1
Nk
!N
n1,...,nk=1
T
n!(m)
m Am−1T
n!(m−1)
m−1 Am−2 . . . A1T
n!(1)
1 ,
where k " m and ! : {1, . . . ,m} #{1, . . . , k} is a surjective map.We show that,
on general Banach spaces and without any restriction on the partition !, the above
averages converge strongly as N # $ under some quite weak compactness
assumptions on the operators Tj and A j . A formula for the limit based on the
spectral analysis of the operators Tj and the continuous version of the result are
presented as well.