Quantum operator hermiticity. Show that S is Hermitian

[engstruggle]

Homework Statement

Spin Operator S has eigenvectors |R> and |L>,
S|R> = |R>
S|L> =-|L>

eigenvectors are orthonormal

Homework Equations

Operator A is Hermitian if <ψ|A|Θ> = <Θ|A|ψ>*

The Attempt at a Solution

<ψ|S|L> = <L|S|ψ>* // Has to be true if S is Hermitian
LHS: <ψ|S|L> = <ψ|-|L>
<ψ|-|L>* = <L|-|ψ>

Question: how do i know how S acts on any function like |ψ> ?
Could somebody provide an algorithm to find if an operator is Hermitian.
I have another example of operator P, where P|R> = |L>
P|L> = |R>
How should i go on about this?

 

[unscientific]

Use the definition of a hermitian operator:

$$\langle\psi|S|\psi\rangle^{\dagger} = \langle\psi|S^{*}|\psi\rangle = \langle\psi|S|\psi\rangle$$