Toeplitz projections and essential commutants

We construct a Toeplitz projection for every regular A  -isometry T∈B(H)nT∈B(H)n on a complex Hilbert space HH and use it to determine the essential commutant of the set of all analytic Toeplitz operators formed with respect to an essentially normal regular A-isometry. We show that the Toeplitz projection annihilates the compact operators if and only if T possesses no joint eigenvalues. As an application we deduce an essential version of the classical Hartman–Wintner spectral inclusion theorem, give a new proof of Johnson and Parrot's theorem on the essential commutant of abelian von Neumann algebras for separable Hilbert spaces and construct short exact sequences of Toeplitz algebras.