How Do You Prove the Adjoint of a Product of Operators?

[mhazelm]

Homework Statement

For operators A, B, prove that (AB)^t = (B)^t(A)^t where ^t is representing the Hermitian adjoint.

I know that this should be similar to proofs I did about matrix transposes in linear algebra, but I'm not sure how to do it without seeing the operators as matrices with indices. I've been trying to do it with dirac notation but that's been confusing...

Homework Equations

The Attempt at a Solution

Well... it's a long shot. I don't think this works:

<(AB)^t psi1| psi2> = ((AB)^t)* < psi1|psi2> = B^t A^t < psi1|psi2> = <psi1| B^tA^t psi2> ==> (AB)^t = B^t A^t.

I think I kind of made it up at the part with the complex conjugate, so yeah, basically I'm confused, even though this is supposed to be the easiest proof ever...

 

[Dick]

<(AB)^T(psi1)|psi2>=<psi1|(AB)(psi2)>=<psi1|A(B(psi2))>=<A^T(psi1)|B(psi2)>. Can you continue?

 

[mhazelm]

aha... I wasn't sure if I was allowed to "split them up", so to speak, at the point where the A moves back to the rhs of the statement. If I'm allowed to do that I'm done!

 

[mhazelm]

oooh wait, it's just composition of functions! hehehe