Bishop-type theorems for non-subnormal operators

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 2 -isometries behave similarly to subnormal operators in the sense that the role of subnormal operators is played by 2 2 -isometries, while the role of normal operators is played by Brownian unitaries. By this analogy, we pose two problems: first, whether the set of all 2 2 -isometries is equal to the strong closure of the set of all Brownian unitaries; second, whether the set of all 2 2 -isometries is strongly closed. In this paper, we give partial solutions to both problems. 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. It turns out that the latter method is applicable only to Brownian unitaries of a fixed covariance. We also investigate the strong and ∗ * -strong closedness of the class of Brownian unitaries.