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?



Addition of essentially self-adjoint operators
Let $D=H^4(0,1)\cap H^2\_0(0,1)$, $Au=u''''$, $Bu=-u''''+u''$. 


