p-Summable Commutators in Dimension d

TL;DR

This paper investigates the properties of invariant subspaces for d-shifts of finite rank, showing that their associated compressed operators generate algebras with commutators in Schatten classes, leading to stability results in operator theory.

Contribution

It introduces new conditions under which compressed d-shift operators generate algebras with controlled commutator properties, extending understanding of their structure and stability.

Findings

Commutators belong to Schatten class L^p for p > d.

The generated C*-algebra is commutative modulo compact operators.

The index formula for the curvature invariant is stable under perturbations.

Abstract

We show that many invariant subspaces M for d-shifts (S_1,...,S_d) of finite rank have the property that the projection P onto M almost commutes with the S_k in the sense that the commutators PS_k - S_kP belong to the Schatten-von Neumann class L^p for every p > d. In such cases the d-tuple of operators (T_1,...,T_d) obtained by compressing (S_1,...,S_d) to the orthocomplement of M generates a *-algebra whose commutator ideal is contained in L^p, p > d. It follows that the C*-algebra generated by T_1,...,T_d is commutative modulo compact operators, the associated Dirac operator is Fredholm, and the index formula for the curvature invariant is stable under compact perturbations and homotopy for this restricted class of d-contractions. We conjecture that the latter conclusions persist under much more general circumstances.