Let $\{Z(t)\}_{t\geq 0}$ be a semigroup of positive operators on a Banach lattice $X$. I want to show that

$$\int_0^1[Z(1-s)+Z(1+s)]fds-f\in X_+,\quad f\in X_+$$

Where $X_+$ denotes the positive part of $X$.

  • $\begingroup$ Can you provide a little more context? Why do you want to show that? Do you have a reason to believe that it's true? Is it homework, research or something else? $\endgroup$ – Joonas Ilmavirta Oct 2 '14 at 7:51
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    $\begingroup$ Try $X$ as the (uniformly) continuous bounded functions on $\mathbb R$, the translation group $Z(t)(f)(x)=f(x+t)$ and a positive function $f$ with $f(0)=1$ which decays rapidly (e.g. $f(x)=\exp(-|x|)$). $\endgroup$ – Jochen Wengenroth Oct 2 '14 at 9:15
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    $\begingroup$ On the first sight the previous example of @JochenWengenroth can be made one-dimensional by taking $X=\mathbb{R}$ and $Z(t)=e^{-t}$. $\endgroup$ – András Bátkai Oct 2 '14 at 9:32
  • $\begingroup$ Thank you so much for these meaningful suggestions but I don't want to be particular. it is a part of my research problem and I want such results for some general space ordered space. since $Z(t)$ are positive operators I do feel it must be satisfied. $\endgroup$ – user786 Oct 2 '14 at 20:04
  • $\begingroup$ But the examples suggest that this cannot be true... $\endgroup$ – András Bátkai Oct 4 '14 at 12:28

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