# Showing that general operator equations are hermitian

#### UCLphysics

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. 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

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#### 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

Science Advisor
Homework Helper
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

Science Advisor
Homework Helper
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.

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