functional calculus

Functional calculus is the art of plugging an operator into a function. Just as you can square a matrix or exponentiate it, functional calculus lets you form f(T) for a whole class of functions f — square roots, exponentials, indicator functions — in a way that is consistent: sums of functions go to sums of operators, products to products, and the constant function 1 goes to the identity. It turns abstract operators into objects you can manipulate like numbers on their spectrum.

The simplest version is polynomial: for p(t) = sum a_k t^k set p(T) = sum a_k T^k. This extends in stages. The holomorphic functional calculus defines f(T) for f analytic on a neighborhood of the spectrum via a Cauchy-type integral of f(lambda) against the resolvent. For a normal (or self-adjoint) operator the continuous and Borel functional calculi go much further, defining f(T) for every continuous, and then every bounded Borel, function on the spectrum, using the projection-valued spectral measure.

The guiding principle, the spectral mapping theorem, says sigma(f(T)) = f(sigma(T)): the spectrum of f(T) is exactly f applied to the spectrum of T. This is why functional calculus is so powerful — it reduces operator questions to ordinary questions about a function restricted to a set of numbers, the spectrum. It is the rigorous machine behind defining sqrt(T) for a positive operator or e^{itT} for a self-adjoint generator of time evolution.