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 or Project Euclid. 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?

  • $\begingroup$ The first two books I pulled from my shelf do unbounded operator theory in the context of Banach spaces. They are the obscure :) books Dunford & Schwartz and Pazy's "Semigroups of linear operators and applications to partial differential equations". I recall that Kato also does much of the theory in Banach spaces. I did not try to check whether D-S or Pazy state the specific facts you noted. I think Pazy usually specializes to the case of reflexive spaces when duality plays an important role. $\endgroup$ – Bill Johnson Apr 26 '12 at 18:06
  • $\begingroup$ @Bill: Thanks for those suggestions. Kato, Chapter III Section 5 gets very close to what I had in mind. $\endgroup$ – Matthew Daws Apr 27 '12 at 14:16

See \S 36 of G. K\"{o}the: Topological Vector Spaces, Vol. 2

  • $\begingroup$ This almost does it in too much detail; but it's a pretty good reference. Thanks! $\endgroup$ – Matthew Daws Apr 27 '12 at 10:50

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