5
$\begingroup$

I am reading Ulrich Krengel's book, Ergodic Theorems; the theorem of Akcoglu's he mentions of is on page 189, theorem 2.5.

" If $T$ is a positive contraction in a space $L_p$ with $1<p<\infty$, then: $$\| \sup_{n\geq 1} A_n |f| \|_p \leq q \| f \|_p$$ "

On page 193, in the notes he mentions that the proof of this theorem for positive power bounded $T$ is still open.

My question, is this problem as of yet still open, and if this indeed the case, what are the attempts at reconciling this open question? any added literature over the years since the publication of Krengel's book?

P.S $$A_n f = A_n(T)f = n^{-1} \sum_{j=0}^{n-1} T^j f $$

$\endgroup$
2
  • $\begingroup$ Pardon the obviousness of this remark, but have you tried emailing Akcoglu or Krengel? $\endgroup$
    – Ian Morris
    Jan 12, 2015 at 14:24
  • $\begingroup$ I emailed Ulrich Krengel enquiring on something else in his book, but didn't get an answer. (more than a month ago). I haven't yet emailed Akcoglu; I'll try to reach him. $\endgroup$
    – Alan
    Jan 13, 2015 at 15:48

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.