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Cosma's NotebooksenOperator Semigroups
http://bactra.org/notebooks/2017/02/27#operator-semigroups
<P>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.
<P>I supposedly learned about operator groups and semigroups when
learned <a href="quantum-mechanics.html">quantum mechanics</a>, but if I'm
honest that didn't make a lot of sense at the time. Things really clicked when
I studied <a href="chaos.html">dynamical systems</a>
and <a href="markov.html">Markov processes</a>. 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.
<P>Actually, there are <em>two</em> 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 <em>functions</em> 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.
<P>I would now like to understand all this more deeply and abstractly.
<ul>Recommended:
<li>Stewart N. Ethier and Thomas G. Kurtz, <cite>Markov Processes: Characterization and Convergence</cite>
<li>Einar Hille, <cite>Functional Analysis and Semi-Groups</cite>
[Actually, I've only read about half of this]
<li>Andrzej Lasota and Michael C. Mackey, <cite>Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics</cite> [Has a really excellent discussion of the Hille-Yosida theorem]
</ul>
<ul>Modesty forbids me to recommend:
<li>CRS, <cite><a href="http://www.stat.cmu.edu/~cshalizi/almost-none/">Almost None of the Theory of Stochastic Processes</a></cite> [I tried to be consistent and clear about presenting Markov process theory from this point of view...]
</ul>
<ul>To read:
<li>Bernd Carl, <cite>Entropy, Compactness, and the Approximation of
Operators</cite>
<li>Adam Bobrowski, <cite><a href="http://cambridge.org/9781107137431">Convergence of One-parameter Operator Semigroups:
In Models of Mathematical Biology and Elsewhere</a></cite>
<li>T. Eisner, B. Farkas, M. Haase and R. Nagel,
<cite><a href="http://www.springer.com/book/9783319168975">Operator Theoretic Aspects of Ergodic Theory</a></cite>
<li>Klaus Jochen Engel, <cite>A Short Course on Operator Semigroups</cite>
<li>Carlos Kubrusly, <cite>Elements of Operator Theory</cite>
<li>Thomas G. Kurtz, "Semigroups of Conditioned Shifts and
Approximation of Markov
Processes", <a href="http://projecteuclid.org/euclid.aop/1176996305"><cite>Annals
of Probability</cite>
<strong>3</strong> (1975): 618--642</a>
</ul>