Linear Graph Transformations and Fiber Dimension with Shift plus Complex Volterra Operator in Invariant Subspaces

Abstract

We study the spectral properties and complemented invariant subspaces in the Bergman spaces having the codimension two property and also with the spectra of some translation and fiber dimension for invariant subspaces. We determine the index of invariant subspaces in Hilbert spaces of vector-valued analytic functions of several complex variables. We classify similarity, reducing manifolds, unitary equivalence of Volterra operators and Volterra invariant subspaces of Hardy spaces. We show nearly invariant subspaces of the backward shift and shift plus complex Volterra operator. We characterize the analytic continuability of Bergman inner functions and linear graph transformations on spaces of analytic functions. We give the algebraic properties of the index of invariant subspaces of operators and of parabolic self-maps on Banach and Hardy spaces.