## Abstract

Commutative-algebraic techniques, together with methods in operator theory and function theory, are used to study the Dirichlet space over the unit disc. For any invariant subspace M ⊂ script D sign ⊗ ℂ^{N} of ℂ^{N}-valued Dirichlet space N - dim(script M sign ⊖ zscript M sign) is exactly the shift Samuel multiplicity of script M sign ^{⊥} = script D sign ⊗ ℂ ⊖ script M sign. The following monotonic property which generalizes a result of Richter is a consequence. If script M sign_{1} ⊂ script M sign_{2} ⊂ script D sign ⊗ ℂ^{N} are two invariant subspaces, then dim(script M sign_{1} ⊖zscript M sign_{1}) ≦ dim(script M sign_{2} ⊖ zscript M sign_{2}). A notion of curvature invariant is introduced for the Dirichlet space, and is shown to be equal to the Fredholm index (up to a sign). A version of the Gauss-Bonnet-Chern formula is also available. A notion of multiplicity is introduced for both invariant and coinvariant subspaces. They are defined by asymptotic formulas and turn out to be integers.

Original language | English |
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Pages (from-to) | 189-211 |

Number of pages | 23 |

Journal | Journal fur die Reine und Angewandte Mathematik |

Issue number | 569 |

DOIs | |

State | Published - 2004 |