Resolve The Transitive Algebra On The Hilbert Space, The Reduced Algebra And Invariant

This thesis mainly concerns transitive algebras, reductive algebras and related invariants on analytic Hilbert spaces.The transitive algebra problem and reductive algebra problem are well-known open problems in operator theory, which had their heyday from 1960s to 1970s. But after that there are few essential progress towards the problems. In this thesis, we consider these two problems by introducing a new tool:fiber dimension. Fiber dimension is an effective invariant in function theory and operator theory. On re-producing analytic Hilbert spaces, we reduce the transitive algebra problem to the calculus of fiber dimension of invariant graph subspaces. This links the transitive algebra problem to the fiber dimension for the first time, which provides a new approach for the study of this problem. Moreover, we obtain some important prop-erties of fiber dimension, such as stability under finite dimensional perturbation, additivity, and so on.Based on these results, we consider the transitive algebra problem and reductive algebra problem on analytic Hilbert spaces with complete Nevanlinna-Pick kernel. It is shown that the two problems have positive answer if the algebras contain all analytic multiplication operators. Furthermore, the corresponding conclusion holds on multiplier invariant subspaces.Lastly we study an additive invariant defined on the invariant subspaces of vector-valued Hardy space over the ball, which is an extension of Arveson curva-ture invariant to a non-Nevanlinna-Pick kernel space. We prove that this invariant equals to the fiber dimension, and equals to Samuel multiplicity under certain condi-tions. For pure isometry with finite multiplicity, we establish an interesting relation between this invariant and Fredholm index.