# Operator Semigroups Simplified

How do I explain Operator Semigroups, in particular, positive operator semigroups to someone who hasn't studied math beyond high school?

I just want to give a vague idea/analogy to someone to let them know a bit about my a project I am working on.

• It depends on what you are doing with them. I don't think just explaining what they are is particularly motivating. – Don Thousand Jan 22 at 18:09
• @DonThousand Right now, I am just studying Positive Operator Semigroups. However, I just want to give someone from a non-math background a rough idea of what I'm studying. How do I do that without getting technical? – Mark Jan 22 at 18:12
• Wikipedia provides a good starting point. – Don Thousand Jan 22 at 18:19
• Were you likely to understand positive operator semigroups when you were in High School? – DisintegratingByParts Jan 23 at 7:30
• @DisintegratingByParts I don't really want them to completely understand. Something like a real-world application or analogy of sorts. Just give a really basic idea. – Mark Jan 23 at 10:12

Maybe, a possible analogy is the exponential function: you are studying a generalization of $$f(t)=e^{a t}$$ which allow ''matrix exponents''.
They are the functional equation $$f(x+y)=f(x)f(y)$$ and the differential equation $$f'(x)=af(x).$$ If we assume that $$f$$ is real-valued, then a solution is the exponential function $$f(t)=e^{at}$$. If we assume that $$f$$ is matrix-valued, then a solution will be given by a ''matrix exponential''. If we want go one step further (which have important applications), we will need semigroup of operators. Here is where your project starts.