Spectral Measures of Unitary Operators

Chapter 5 builds a rigorous, implementable theory for computing spectral measures of unitary operators. It mirrors Chapter 4, but there are key differences. For eigenvalue-based methods, discretisations must be unitary. We show how to enforce this via polar decomposition, and that only weak-operator convergence is needed for the resulting measures to converge. On the resolvent side, Stone-type formulas on the circle and the Carathéodory function lead to Poisson-kernel smoothing computed by solving linear systems. To reduce cost and sharpen resolution, we introduce higher-order periodic kernels—rational resolvent kernels and trigonometric-polynomial Fourier filters—that accelerate convergence. A change of variables then transfers the machinery to bounded self-adjoint operators, yielding an infinite-dimensional convergence theory for the kernel polynomial method from Chebyshev moments. The techniques are highly flexible and illustrated on challenging examples, including singular spectral measures and the unitary almost Mathieu operator.