# Journal of Operator Theory

Volume 78, Issue 1,  Summer  2017  pp. 21-43.

Cowen-Douglas tuples and fiber dimensions

Authors:  Jorg Eschmeier (1) and Sebastian Langendorfer (2)
Author institution: (1) FR Mathematik, Universitat des Saarlandes, Saarbrucken, 66041, Germany
(2) FR Mathematik, Universitat des Saarlandes, Saarbrucken, 66041, Germany

Summary:  Let $T \in L(X)^n$ be a Cowen--Douglas tuple on a Banach space $X$. We use functional representations of $T$ to associate with each $T$-invariant subspace $Y\subset X$ an integer called the fiber dimension $\mathrm{fd}(Y)$ 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.

DOI: http://dx.doi.org/10.7900/jot.2016may04.2134
Keywords:  Cowen-Douglas tuples, fiber dimension, Samuel multiplicity, holomorphic model spaces