Hermitian conjugate of operators

[v_pino]

Homework Statement

Find the hermitian conjugates, where A and B are operators.

a.) AB-BA
b.) AB+BA
c.) i(AB+BA)
d.) $A^\dagger A$

Homework Equations

$(AB)^\dagger =B^\dagger A^\dagger$

The Attempt at a Solution

Are they correct and can I simplify them more?

a.) $(AB-BA)^\dagger = (AB)^\dagger - (BA)^\dagger= B^\dagger A^\dagger - A^\dagger B^\dagger = BA-AB = -(AB-BA)$

b.) $(AB+BA)^\dagger = (AB)^\dagger + (BA)^\dagger= B^\dagger A^\dagger + A^\dagger B^\dagger = BA+AB$

c.) $[i(AB+BA)]^\dagger = -i[(AB)^\dagger + (BA)^\dagger]= -i(B^\dagger A^\dagger + A^\dagger B^\dagger) = BA+AB$

d.) $[A^\dagger A]^\dagger = A^\dagger (A^\dagger)^\dagger=A^\dagger A$

 

[dextercioby]

You might review point c.)

 

[v_pino]

For point C.) $-i(B^\dagger A^\dagger + A^\dagger B^\dagger)$

 

[dextercioby]

Alright. The last equality is not true. That's what I meant.

 

[v_pino]

Is it ok until $-i(B^\dagger A^\dagger + A^\dagger B^\dagger)$?

 

[dextercioby]

Yes, it is.