During my PhD, I further developed the spectral theory of non-normal operators. I introduced the meromorphic functional calculus and new results for spectral projection operators for both greater generality and ease of use. (See my “Beyond the spectral theorem“ paper.) Much of my work leverages this spectral theory in one form or another.
Non-normal spectral theory is extremely powerful! Even nonlinear dynamics are linear on a larger space spanned by observables. The time-evolution operator is generically non-normal though——it doesn’t commute with its Hermitian adjoint. Transition matrices of Markov or hidden Markov models are prime examples of non-normal operators.
