Those concepts were introduced, i.e.
Adjoint Operator
It's an operator $T^{*} \: : \: H_2 \rightarrow H_1$ such that $\forall _{x\in H_1,\:y\in H_2} \: \langle Tx, y \rangle_2 = \langle x, T^{*} y \rangle_1$
Dual Operator
We call $T'$ a dual operator, if: $$T' : \: Y^{*} \ni \Psi \: \rightarrow \: T' \Psi \in X^{*}$$ where $X^{*} = \mathcal{B}(X,\mathbb{C})$, $Y^{*} = \mathcal{B}(Y,\mathbb{C})$
So the definitions are straight forward, but why are we introducing them in the first place? What's their "practical" use in Mathematics? And what's the intuition behind them?
Also, why is it important/interesting for them to be compact?