Inner functions and zero sets for ${\mathrm{\ell }}_A^p$

In this paper we characterize the zero sets of functions from ${\mathrm{\ell }}_A^p$ (the analytic functions on the open unit disk $D$ whose Taylor coefficients form an ${\mathrm{\ell }}^p$ sequence) by developing a concept of an “inner function” modeled by Beurling’s discussion of the Hilbert space ${\mathrm{\ell }}_A^2$ (the classical Hardy space). The zero set criterion is used to construct families of zero sets which are not covered by classical results. In particular, it is proved that when $p>2$, there are zero sets for ${\mathrm{\ell }}_A^p$ which are not Blaschke sequences.