Operator Semigroups

25 Jan 2016 16:50

In math, an "operator" is just a mapping which takes points in a function space to other points in another function space; the term is used even or especially when the two spaces are the same, which is what I'm interested in here. (Taking derivatives, integrals and Fourier transforms are all familiar examples.) An "operator semigroup" is, naturally, a collection of operators which forms a semigroup, raising the question of what the latter term means. Here it means that when we compose two operators from the collection, we get another operator in the collection, i.e., that when $ A $ and $ B $ are in the semigroup, so is $ AB $; and that composition is associative, so that $ (AB)C = A(B) $. If one of the operators is the identity, then the semigroup is sometimes called a "monoid". The semigroup becomes a group if every operator has an inverse, which is not the case for many natural examples.

I supposedly learned about operator groups and semigroups when learned quantum mechanics, but if I'm honest that didn't make a lot of sense at the time. Things really clicked when I studied dynamical systems and Markov processes. For discrete-time dynamical systems, the operator semi-group is just the powers of the time-evolution operator, a.k.a. the Frobenius-Perron (or Perron-Frobenius) operator; for discrete-time Markov chains, the powers of the transition matrix. In continuous time, one has the more subtle notion of a generator, and the Hille-Yosida theorem linking generators to semigroups indexed by a single continuous parameter.

Actually, there are two families of semigroups for dynamical systems and Markov processes. One describes the evolution of individual points or probability measures under the dynamics. The other describes the conditional expectation of functions over the state space. (For dynamical systems, this is called the "Koopman operator".) These correspond, in quantum mechanics, to the Schrödinger and Heisenberg pictures, respectively. This is related to the duality between measures and integrable functions --- integrating a function with respect to a measure gives you a single real number, so you can think of measures as one-forms on the vector space of functions.

I would now like to understand all this more deeply and abstractly.