Fiber dimension for invariant subspaces

In this paper we study the fiber dimension of invariant subspaces for a large class of operators. We define a class of invariant subspaces called CF subspaces which are related to the codimension-one property. We obtain several characterizations of CF subspaces, including one in terms of Samuel multiplicity.Other new findings include: (1) a lattice-additive formula and its applications (Section 4); (2) a new concept of “absorbance” which describes a rough containment relation for invariant subspaces (Section 5); (3) the existence of a unique, smallest CF subspace containing an arbitrary invariant subspace and preserving the fiber dimension (Section 6).