The spectral theory of Toeplitz operators applied to approximation problems in Hilbert spaces

The solution to a particular constrained approximation problem, in an abstract Hilbert space setting, may be interpreted in terms of a generalised Toeplitz operator. We consider concrete versions of this problem, in settings which involve generalised Hardy spaces, Paley-Wiener spaces and the Segal-Bargmann space, and derive spectral representations of the associated Toeplitz operators.