Cyclic row contractions and rigidity of invariant subspaces

TL;DR

This paper studies cyclic pure row contractions, showing they can be transformed into standard shift compressions, and explores invariant subspace structures and decompositions, especially for nilpotent cases, revealing algebraic and function-theoretic properties.

Contribution

It extends classification results of pure row contractions by relaxing defect conditions to cyclic vectors and analyzes invariant subspace decompositions, especially for nilpotent contractions.

Findings

Cyclic pure row contractions can be transformed into standard shift compressions.

Invariant subspace decompositions face algebraic obstacles in multivariate settings.

Nilpotent commuting row contractions exhibit specific rigidity properties in their invariant subspaces.

Abstract

It is known that pure row contractions with one-dimensional defect spaces can be classified up to unitary equivalence by compressions of the standard $d$-shift acting on the full Fock space. Upon settling for a softer relation than unitary equivalence, we relax the defect condition and simply require the row contraction to admit a cyclic vector. We show that cyclic pure row contractions can be "transformed" (in a precise technical sense) into compressions of the standard $d$-shift. Cyclic decompositions of the underlying Hilbert spaces are the natural tool to extend this fact to higher multiplicities. We show that such decompositions face multivariate obstacles of an algebraic nature. Nevertheless, some decompositions are obtained for nilpotent commuting row contractions by analyzing function theoretic rigidity properties of their invariant subspaces.