A nonprincipal invariant subspace of the Hardy space on the torus

Abstract:

Let $H^2(U^n)$ be the usual Hardy space (with index 2) of holomorphic functions on $U^n$, the unit polydisc in complex $n$-space. A subspace of $H^2(U^n)$ is invariant if closed under multiplication by the coordinate functions. To solve a problem left open in a paper of P. R. Ahem and D. N. Clark and a book by W. Rudin the author constructs a closed invariant subspace $M$ of $H^2(U^2)$ with (1) an $f$ in $M$ never vanishing on $U^2$ and (2) each $g$ in $M$ being contained in a proper closed invariant subspace of $M$. This easily extends to $n \geqq 2$.