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Hermitian Operators Help

Basically I've seen some expressions involving Hermitian Operators that I can't seem to justify, that others on the internet throw around like axiomatic starting points.

(AB+BA)+ = (AB)++(BA)+? Why does this work?

Assuming A&B are hermitian, I get why we can assume A+B is hermitian, but does it follow that A-B is hermitian? Because AB is only hermitian if AB=BA which means AB-BA=0, and I'm fairly sure 0 cannot be a hermitian operator.

About the simplest way to see it is to work out (AB)+ + (BA)+ to get B+A+ + A+B+ then involution identity property to further simplify. Also, the zero matrix is its own Hermitian conjugate; has real eigenvalues, all zero.

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