# Domain of essential self-adjointness for $A\otimes 1+1\otimes A$

Let $$A$$ be an unbounded self-adjoint operator acting on a Hilbert space $$H$$ (typically $$L^2(\mathbb{R}^d)$$).

Then, using Stone's theorem, the operator $$A^{\otimes 2}:=A\otimes 1+1\otimes A$$ defines a self-adjoint operator on $$H\otimes H$$, the domain of which might be difficult to determine.

Is it however true that if $$A$$ is furthermore essentially self-adjoint on $$\mathcal{C}$$, then $$A^{\otimes 2}$$ is essentially self-adjoint on $$\mathcal{C}\otimes \mathcal{C}$$ ?

This is true and in fact even more is true: it is proved in

Theorem VIII.33 (a) of Reed & Simon, Methods of Modern Mathematical Physics I: Functional Analysis, 1980

that

If $$\{A_k : \mathcal{H}_k \to \mathcal{H}_k \}$$, $$1 \leq k \leq N$$, is a collection of self-adjoint operators with $$D^e_k \subset \mathcal{H}_k$$ the domain of essential self-adjointness for $$A_k$$ and $$P \in \mathbb{R}[x_1, \ldots, x_N]$$ is of degree $$n_k$$ in the variable $$x_k$$, then $$P(A_1^{(1)}, \ldots, A_N^{(N)})$$ is essentially self-adjoint on $$D^e := \bigotimes_{k=1}^N D^e_k$$,

wherein I've used the notation $$A_k^{(k)} := I \otimes \cdots \otimes A_k \otimes \cdots I$$ (which in the text is written just as $$A_k$$, overloading the notation). The specialization to $$\sum_{k=1}^N A_k^{(k)}$$, which is only slightly more general than your question, is then

Corollary (a) to Theorem VIII.33, ibid.

The proof to your specific form would involve, I believe, essentially the same ingredients as that to Theorem VIII.33 (a). The natural consequence for the closure of the spectrum is also dealt with in the same place.

• Thank you very much for your help. R&S contains so much information, i will include this reference.
– Chr
Commented Apr 19, 2020 at 8:32