Approximate spectral cosynthesis in the Harmonically weighted Dirichlet spaces

Abstract

For a finite positive Borel measure μ on the unit circle, let D(μ) be the associated harmonically weighted Dirichlet space. A shift invariant subspace M recognizes strong approximate spectral cosynthesis if there exists a sequence of shift invariant subspaces M_k, with finite codimension, such that the orthogonal projections onto M_k converge in the strong operator topology to the orthogonal projection onto M. If μ is a finite sum of atoms, then we show that shift invariant subspaces of D(μ) admits strong approximate spectral cosynthesis.