On the topology of invariant subspaces of a shift of higher multiplicity

TL;DR

This paper generalizes the topological classification of invariant subspaces from the shift of multiplicity one to arbitrary finite multiplicity, establishing a correspondence with the space of inner functions.

Contribution

It extends the known results on invariant subspaces of the shift operator from multiplicity one to higher finite multiplicities, providing a new topological framework.

Findings

Established a one-to-one correspondence between invariant subspaces and inner functions for finite multiplicity shifts.

Generalized the topological description of invariant subspace lattices beyond multiplicity one.

Connected the structure of invariant subspaces with the space of inner functions for higher multiplicities.

Abstract

Following Beurling's theorem and a study of the topology of invariant subspaces by R. Douglas and C. Pearcy description of path connected components of invariant subspace lattice for shift of multiplicity one has been given by R.Yang. This paper generalizes result to arbitrary finite multiplicity. We show that there exists one to one correspondence between the invariant subspace lattice of shift of arbitrary finite multiplicity and the space of inner functions.