The Agler-Young Class

TL;DR

This paper introduces the Agler-Young class of operator tuples, providing a Wold decomposition, dilation theorem, and characterizations, extending classical results to non-commutative operator settings with applications to Toeplitz operators.

Contribution

It defines the Agler-Young class and establishes a Wold decomposition, dilation theorem, and hereditary functional calculus characterization, extending known results to non-commutative operators.

Findings

Wold decomposition for the Agler-Young class

Dilation theorem with explicit structure

Characterization via hereditary functional calculus

Abstract

This note introduces a special class of tuples of bounded operators on a Hilbert space. It is called the Agler Young class. Major results about this class include a Wold decomposition and a dilation theorem. The structure of the dilation is completely spelt out. A characterization of this class using the hereditary functional calculus of Agler is obtained and examples are discussed. Toeplitz operators play a major role in this note. An Agler-Young pair arising from a truncated Toeplitz operator is characterized. Thus, we extend results obtained in the case of commuting operators by several authors over many decades to the non-commutative situation. The results for the commuting case can be recovered as special cases.