Why does (AB+BA)+ = (AB)+ + (BA)+ for Hermitian operators?

[danmel413]

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.

Thanks

 

[SrayD]

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.