# Real and imaginary part of an unbounded normal operator

It is well-known that any bounded normal operator $$A$$ can be written as $$A= A_1+ i A_2$$ where $$A_1$$ and $$A_2$$ are commuting bounded self-adjoint operators. This leads to a proof of spectral theorem of bounded normal operators using spectral theorem of bounded self-adjoint operators.

My question is, for an unbounded normal operator $$A$$, is there also such a decompostion $$A=A_1+i A_2$$ with $$A_1$$, $$A_2$$ (unbounded) self-adjoint? If so, is there an elementary proof of this without using spectral theorem of unbounded normal operators?

• Just to remind for unbounded operator, normal means closed and $AA^*=A^*A$ and self-adjoint means $A=A^*$(in the sense that their domain coincides), not merely symmetric – Yuchen Liao Oct 7 '18 at 15:37