
Multiplication Invariant Subspaces of Hardy Spaces
This paper studies closed subspaces $L$
of the Hardy spaces $H^p$ which are $g$invariant ({\it i.e.},
$g\cdot L \subseteq L)$ where $g$ is inner, $g\neq 1$. If
$p=2$, the Wold decomposition theorem implies that there is
a countable ``$g$basis'' $f_1, f_2,\ldots$ of
$L$ in the sense that $L$ is a direct sum of spaces
$f_j\cdot H^2[g]$ where $H^2[g] = \{f\circ g \mid f\in H^2\}$.
The basis elements $f_j$ satisfy the
additional property that $\int_T f_j^2 g^k=0$,
$k=1,2,\ldots\,.$ We call such functions $g$$2$inner.
It also
follows that any $f\in H^2$ can be factored $f=h_{f,2}\cdot
(F_2\circ g)$ where $h_{f,2}$ is $g$$2$inner and $F$ is
outer, generalizing the classical Riesz factorization.
Using $L^p$ estimates for the canonical decomposition of
$H^2$, we find a factorization $f=h_{f,p} \cdot (F_p \circ
g)$ for $f\in H^p$. If $p\geq 1$ and $g$ is a finite
Blaschke product we obtain, for any $g$invariant
$L\subseteq H^p$, a finite $g$basis of $g$$p$inner
functions.
Categories:30H05, 46E15, 47B38 