My initial impression is that for what you want, you're going to need a notion of $A^*: E\to E$ when $A:E\to E$ is an operator on a Banach space. I don't know much about this, but some years ago did see this short paper
MR2053349 (2005a:46045)
Gill, Tepper L.(1-HWRD-EE); Basu, Sudeshna(1-HWRD); Zachary, Woodford W.(1-HWRD-EE); Steadman, V.(1-DC)
Adjoint for operators in Banach spaces. Proc. Amer. Math. Soc. 132 (2004), no. 5, 1429--1434
which requires a choice of Hilbert space rigging $H_1 \hookrightarrow E \hookrightarrow H_2$.
One thing that might go wrong with $(1) \implies (3)$ in general Banach spaces is the non-existence, in general, of projections from $E$ onto a closed subspace. However, that doesn't rule out the possiblity that something like $(1)\implies(3)$ does indeed hold; I'd need to think about this a bit more.