  Let T be a Cowen-Douglas tuple on a Banach space X. We use functional
representations of T to associate with each T-invariant subspace Y of X an
integer called the fiber dimension of Y. Among other results we prove a limit
formula for the fiber dimension, show that it is invariant under suitable
changes of Y and deduce a dimension formula for pairs of homogeneous invariant
subspaces of graded Cowen-Douglas tuples on Hilbert spaces. We thus extend
results proved by L. Chen, G. Cheng and X. Fang for single Cowen-Douglas
operators on Hilbert spaces to the case of commuting operator systems on Banach
spaces.
