Approximate spectral cosynthesis in the harmonically weighted Dirichlet spaces

Abstract

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

Keywords

• weighted Dirichlet spaces
• invariant subspaces
• strong approximate spectral cosynthesis

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