There's well-known Chebyshev's sum inequality. I'm looking for possible generalization of this statement; problem is that I'm not sure what is the right direction, but here's one which I'd prefer.
Let $A$ be some $\Bbb R$- or $\Bbb C$-algebra with seminorm $|\cdot|$ and "positive cone" $P$.
$T$ is some operator (or parametric family $T_{\alpha}$) on $A$ satisfying several unknown properties; maybe, it looks like a shift operator on $l^1(\Bbb Z)$ or parametric shift on $l^1(R)$.
Template of theorem (Generalized Chebyshev's sum inequality)
If $a, b \in A$ are elements such that $(T-1)a$ and $(T-1)b$ are positive, then $|ab| \geq |a||b|$
If $x, y \in A$ are elements such that $(T-1)x$ and $(1-T)y$ are positive, then $|xy| \leq |x||y|$
Original inequality is obtained by taking $A = l^1(\Bbb Z), T$ being equal to shift, $P$ usual positive cone in $C^*$ sense and seminorm being stadard or $|x|_n := |x \cdot \chi_{[0, n]}|$. With obvious tweaks one can adjust this to integral version on something like $L^1(\Bbb R)$ with limiting positivity condition for small forward difference operators.
Question 1. What's so special about shift operator?
Question 2. Is there some statement fitting the template and where $T$ does not look like a shift at all?