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 $X$.
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$\begingroup$ 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 IlmavirtaCommented Oct 2, 2014 at 7:51
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1$\begingroup$ 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 WengenrothCommented Oct 2, 2014 at 9:15
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2$\begingroup$ 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 BátkaiCommented Oct 2, 2014 at 9:32
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$\begingroup$ 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. $\endgroup$– user786Commented Oct 2, 2014 at 20:04
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$\begingroup$ But the examples suggest that this cannot be true... $\endgroup$– András BátkaiCommented Oct 4, 2014 at 12:28
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