  We introduce a new concept of frame operators for Banach spaces we call a
Hilbert-Schauder frame operator. This is a hybird between standard frame theory
for Hilbert spaces and Schauder frame theory for Banach spaces. Most of our
results involve basic structure properties of the Hilbert-Schauder frame
operator. Examples of Hilbert-Schauder frames include standard Hilbert frames
and classical bases of $\ell_p$ and $L^p$-spaces with $1< p \le 2$. Finally, we
give a new isomorphic characterization of Hilbert spaces.
