Decomposition into Invariant Subspaces

Abstract

Beginning with this Lecture we will use the invariant subspaces of the shift operator S for the analysis of the spectral properties of the compressions

$${T_\Theta }^{\underline{\underline {def}} }{P_\Theta }S\left| {K, {P_\Theta }} \right. = {P_{{K_\Theta }}}, {K_\Theta } = {H^2}\Theta \Theta {H^2}$$

of this operator onto a coinvariant subspace K_Θ. In this Lecture we only manage to take the first preliminary step in this direction. We try to decompose the operator T_Θ into a sum of simpler parts, connecting such a decomposition with a refinement of the spectrum σ(T_Θ). When a further refinement of the spectrum is not any longer possible, it will be necessary to single out a chain of invariant subspaces, which leads to integral representations for the operator which are analogous to the triangular form for matrices in linear algebra. It turns however out to be most difficult to reconstruct the properties of the operator subject to our investigation from the corresponding properties of its restrictions to the terms of the decomposition. Several of the following lectures will be devoted to this reconstruction. Here we just study the simplest problems connected with spectral subspaces and with finite dimensional subspaces.