Welcome to MathOverflow

Question

0

Say $A, B : H \supset D \to H$ are essentially self-adjoint operators on the dense common domain $D$. $H$ is some Hilbert space. Does it hold that $A + B$ is also essentially self-adjoint? If not, can you please give me a counterexample?

flag	asked Jun 4 2011 at 6:35 jsb 181●3

2

Why don't you just leaf through one of the hundreds of books on spectral theory and/or theory of operators (e.g., Reed and Simon's treatise)? a large part of the theory is devoted to answer precisely this question – Piero D'Ancona Jun 4 2011 at 8:57

Michael Renardy · Accepted Answer · 2011-06-04 07:34:52Z UTC

5

Let $D=H^4(0,1)\cap H^2_0(0,1)$, $Au=u''''$, $Bu=-u''''+u''$.

answered Jun 4 2011 at 7:34

Michael Renardy
4,771●1●10●17

2

It may be helpful to note that the reason counterexamples exist is that the intersection of the two domains may be smaller than each of the domains; and if $A$ is ess. self-adjoint with domain $D$ and $D_0\subset D$ is dense, the closure of the restriction of $A$ to $D_0$ may not coincide with $A$. (If the closure of the restriction of $A$ to $D_0$ is $A$ one sometimes says that $D_0$ is a core for $A$). So your $A+B$ will be ess. self-adjoint if you assume that there is a common core for $A$ and $B$. – Dima Shlyakhtenko Jun 4 2011 at 7:43

In the example above D is a core for A and B. It is not a core for A+B. – Michael Renardy Jun 4 2011 at 10:41

Addition of essentially self-adjoint operators

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.