Invariant subspaces of weighted Bergman spaces in infinitely many variables

This paper is concerned with polynomially generated multiplier invariant subspaces of the weighted Bergman space $A_{\mathit{\beta }}^2$ in infinitely many variables. We completely classify these invariant subspaces under the unitary equivalence. Our results not only cover cases of both the Hardy space $H^2({\mathbb{D}}_2^{\mathrm{\infty }})$ and the Bergman space $A^2({\mathbb{D}}_2^{\mathrm{\infty }})$ in infinitely many variables, but also apply in finite-variable setting.