Cowen-Douglas tuples and fiber dimensions

TL;DR

This paper introduces the concept of fiber dimension for invariant subspaces of Cowen-Douglas tuples on Banach spaces, extending existing results from single operators to operator systems.

Contribution

It develops a functional representation approach to define fiber dimension, proves a limit formula, and generalizes dimension results to commuting operator systems on Banach spaces.

Findings

Established a limit formula for fiber dimension.

Proved invariance of fiber dimension under certain subspace changes.

Extended dimension formulas to pairs of homogeneous invariant subspaces.

Abstract

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.