Showing that general operator equations are hermitian

[UCLphysics]

Homework Statement

a)For a general operator A, show that and i(A-A⁺) are hermitian?

b) If operators A and B are hermitian, show that the operator (A+B)^n is Hermitian.

Homework Equations

The Attempt at a Solution

The first part I did,

(A+A⁺)⁺=(A⁺+A)=(A+A⁺)

i(A-A⁺)=[i(A-A⁺)]⁺=(iA)⁺-(iA⁺)⁺)=i(A-A⁺)

The second part I used binomial expansion (induction) but I was told they do not commute so this cannot be done?

Is my answer to the first part correct and what route should I take for the second? Thanks

 

[Oxvillian]

First part looks about right.

For the second part, do you really need to multiply things out? After all, if A and B are Hermitian then A+B is also Hermitian...

 

[dextercioby]

The problems seems to have no connection with <Advanced Physics> and neither with mathematics, since it's not mathematically (i.e. in agreement with mathematics) formulated and not even semantically.

 

[dextercioby]

For example, the mathematical problem for point a) should have been

"Let A be a densly defined linear operator on a complex separable Hilbert space $\mathcal{H}$.

Prove that if $D(A)\cap D\left(A^{\dagger}\right)$ is dense everywhere in $\mathcal{H}$,

then the operator $i\left(A-A^{\dagger}\right)$ is symmetric.