MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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. – Bill Johnson Apr 26 '12 at 18:06
@Bill: Thanks for those suggestions. Kato, Chapter III Section 5 gets very close to what I had in mind. – Matthew Daws Apr 27 '12 at 14:16
up vote 2 down vote accepted

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

share|cite|improve this answer
This almost does it in too much detail; but it's a pretty good reference. Thanks! – Matthew Daws Apr 27 '12 at 10:50

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.