The Defect Sequence for Contractive Tuples

TL;DR

This paper introduces the defect sequence for contractive tuples of Hilbert space operators, explores its properties, bounds, and special cases like maximal and pure tuples, advancing the understanding of operator theory.

Contribution

It defines the defect sequence for contractive tuples, establishes bounds, and characterizes maximal tuples in both non-commutative and commutative settings.

Findings

Upper bounds for defect dimensions are established.

Creation operators on full Fock space are maximal.

Coordinate multipliers on Drury-Arveson space are maximal.

Abstract

We introduce the defect sequence for a contractive tuple of Hilbert space operators and investigate its properties. The defect sequence is a sequence of numbers, called defect dimensions associated with a contractive tuple. We show that there are upper bounds for the defect dimensions. The tuples for which these upper bounds are obtained, are called maximal contractive tuples. The upper bounds are different in the non-commutative and in the com- mutative case. We show that the creation operators on the full Fock space and the co ordinate multipliers on the Drury-Arveson space are maximal. We also study pure tuples and see how the defect dimensions play a role in their irreducibility.