1
$\begingroup$

I am working with self-adjoint unbounded linear operators on infinite-dimensional separable Hilbert spaces, and I would like to know some references regarding analytic vectors, especially with respect to their properties as subset of the Hilbert space on which they are defined. Moreover I would like to know if there are particular conditions under which the restriction of a self-adjoint, unbounded linear operator to the space of its analytical vector is closed. Thank You.

EDIT

In view of Yurii Savchuk's reply, I understand that the restriction $(A_{r}\,;\mathcal{D}^{\omega}(A))$ of a self-adjoint linear operator $(A\,;\mathcal{D}(A))$ to the set of its analytic vectors, is closed if and only if $A$ is defined over the whole Hilbert space. Therefore, another question arises.

Let $(A\,;\mathcal{D}(A))$ be an unbounded, self-adjoint linear operator. Define:

$$ D^{\infty}(A)=\bigcap_{k=1}^{\infty}(A^{k}) $$ and let $\mathcal{D}(C)$ be a dense subspace of $\mathcal{D}(A)$ which is invariant with respect to $A$. It is true that $\mathcal{D}^{\infty}(A)\subseteq\mathcal{D}(C)$?

$\endgroup$
1
$\begingroup$

Here are some references:

  1. Reed, Simon, "Methods of modern mathematical physics. II. Fourier analysis, self-adjointness."

  2. Schmüdgen, Konrad "Unbounded self-adjoint operators on Hilbert space."

What kind of properties do you expect from analytical vectors?

If the restriction of a self-adjoint operator $(A,D(A))$ to the space of its analytical vectors $D^\omega(A)$ is closed then $(A,D(A))$ is a self-adjoint extention of a self-adjoint operator $(A,D^\omega(A)).$ Hence $D(A)=D^\omega(A).$ The latter is true only for bounded operators. (Proof uses the spectral theorem.)

$\endgroup$
1
$\begingroup$

A counterexample for your EDIT question is $i\frac{d}{dx}$ on $\mathcal{L}^2(\mathbb R)$ with $\mathcal{D}(C) =$ finite linear combinations of $\mathcal{C^\infty} $ functions of compact support.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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