spectral mapping theorem

If the functional calculus tells you how to build f(T), the spectral mapping theorem tells you what its eigenvalues are — and the answer is exactly what you would hope. The spectrum (the set of eigenvalues) of f(T) is the image of the spectrum of T under f. Written compactly: spectrum(f(T)) = f(spectrum(T)). Functions move eigenvalues around in the simplest possible way, applying f one point at a time.

This is almost obvious from the spectral resolution but worth stating because it short-circuits a lot of work. If T has eigenvalues 2 and -3, then T^2 has eigenvalues 4 and 9, exp(T) has eigenvalues exp(2) and exp(-3), and (T - I)^-1 has eigenvalues 1 and -1/4. You never have to recompute a characteristic polynomial; you just push the old eigenvalues through f. The eigenvectors do not move at all, only their labels.

The theorem also delivers clean structural consequences. T is invertible exactly when 0 is not in its spectrum, which is why f(x) = 1/x is allowed precisely then. The spectral radius (largest |eigenvalue|) of T^k is the k-th power of the spectral radius of T. And eigenvalue-based criteria transfer: a self-adjoint T is positive iff its spectrum lies in [0, infinity), so its square root sqrt(T) has spectrum sqrt of those, again nonnegative. In finite dimensions the proof is one line from f(T) = sum f(lambda_i) P_i; the same statement carries over, with care, to bounded operators on infinite-dimensional spaces.