Bishop-like theorems for non-subnormal operators

TL;DR

This paper extends Bishop's theorem to 2-isometries, showing they can be approximated by Brownian unitaries, and explores their operator class properties.

Contribution

It establishes Bishop-like theorems for 2-isometries using two different methods and analyzes the closedness of Brownian unitaries.

Findings

2-isometries can be approximated by Brownian unitaries

Bishop-like theorems are valid for 2-isometries

The class of Brownian unitaries is strongly closed

Abstract

The celebrated Bishop theorem states that an operator is subnormal if and only if it is the strong limit of a net (or a sequence) of normal operators. By the Agler-Stankus theorem, $2$-isometries behave similarly to subnormal operator in the sense that the role of subnormal operators is played by $2$-isometries, while the role of normal operators is played by Brownian unitaries. In this paper we give Bishop-like theorems for $2$-isometries. Two methods are involved, the first of which goes back to Bishop's original idea and the second refers to Conway and Hadwin's result of general nature. We also investigate the strong and $*$-strong closedness of the class of Brownian unitaries.