In the paper "A CRITERION FOR THE NORMALITY OF UNBOUNDED OPERATORS AND APPLICATIONS TO SELF-ADJOINTNESS" by M. H MORTAD (http://arxiv.org/pdf/1301.0241.pdf), the author states the following theorem
Let $A$ and $B$ be two operators with domains $D(A)$ and $D(B)$ respectively, such that $A+B$ is densely defined. Then $A+B$ is
self-adjoint on $D(A)$ if $A$ and $B$ are selfadjoint and if B is bounded.
self-adjoint on $D(A) \cap D(B)$ whenever $A$ and $B$ are commuting selfadjoint and positive operators.
self-adjoint on $D(A) \cap D(B)$ whenever $A$ and $B$ are anti-commuting selfadjoint operators.
self-adjoint on $D(A)$ if $B$ is symmteric and $A$-bounded with relative bounded $a<1$, and $A$ is selfadjoint (Kato-Rellich).
My question concerns point 2. In this point the author refers the reader to C. Putnam, Commutation Properties of Hilbert Space Operators, Springer, 1967. I didn't find this theorem in that book, so if somebody knew that it is there for sure I would be really grateful for giving me more precise information about the location of that theorem. I'm curious why the positivity assumption is important there, in 3. for anti-commuting operators no positivity assumption was made.