Thank you for reading.


My question was raised up when I tried to prove an example in the book of Liggett(1985), which is in P13 Example 2.3(a).


Here is a link of the page:


<https://books.google.com/books?id=7JbqBwAAQBAJ&lpg=PR3&dq=liggett%201985&hl=zh-CN&pg=PA13#v=onepage&q=liggett%201985&f=false>


My questions are as follows:


1) what the definition of positive operator is while the book didn't mention that there is a Hilbert space;


2) how to prove Example 2.3 (a).



A question about positive operator pregenerator
This question is not research level and would be better suited on MathStackExchange. 


Positivity here just means $f\ge 0$ $\Rightarrow$ $Tf\ge 0$ (where $f\ge 0$ is defined as $f(x)\ge 0$ for each $x\in X$). 


The example is verified using Proposition 2.2: If $f(\eta)=\min f(X)$ then $g=f-f(\eta)1\ge 0$ and since $T(1)=1$ we get
$$ (T-I)(f)(\eta)=T(f)(\eta)-f(\eta)=T(f-f(\eta)1)(\eta)\ge 0.$$ 


