Bishop-like 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, $2$-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.