Usually we would define a "densely defined, closed operator" on a Banach space $E$ to be a linear map $T:D(T)\rightarrow E$, where $D(T)$ is a dense subspace of $E$, and the graph of $T$, $G(T)=\{ (x,T(x)) : x\in D(T) \}$ is closed in $E\times E$. Then we can define an adjoint by setting
`\[ D(T^*) = \{ f\in E^* : \exists g\in E^*, f(Tx) = g(x) \ (x\in D(T)) \}. \]`
That $D(T)$ is dense means that if $f\in D(T^\*)$ then the associated $g$ is unique, so we can define $T^\*(f)=g$. This level of generality seems rare-- e.g. Davies in his book "One-parameter semigroups" mentions this, notes that $D(T^\*)$ can fail to be norm dense, and moves on to Hilbert spaces.


Indeed, most books seem to just start out working with Hilbert spaces (and then usually $T^\*$ means the Hilbert space adjoint-- but this is essentially the same thing, up to twisting by some conjugation). Here you can apply Hilbert space techniques to show that $D(T^\*)$ is dense etc.


It seems to me however that $D(T^\*)$ will always at least be weak$^\*$-dense and that $G(T^\*)$ will be weak$^\*$-closed in $E^\*\times E^\*$. Moreover, the proofs don't seem to need Hilbert space techniques. Moreover, starting with such a "weak$^\*$-closed, densely defined operator" on $E^\*$, we can always find a densely-defined closed operator on $E$ which induces it. Applied to a reflexive Banach space, one builds a very satisfactory theory.


The only source I know which talks about "closed" operators in such generality is a paper by Ciorănescu and Zsidó, see [MathSciNet](http://www.ams.org/mathscinet-getitem?mr=0430867) or [Project Euclid](http://projecteuclid.org/euclid.tmj/1178240775). Even they don't mention the duality result.



> 
> My question: Is there a good (or even bad) reference for all this? In particular, that a weak$^\*$-closed operator is the adjoint of a closed operator?
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Closed operators and duality
See \S 36 of 
G. K\"{o}the: Topological Vector Spaces, Vol. 2


