The following portion is paraphrased from Mathematical Methods for Physics and Engineering by Riley, Hobson, and Bence.

• The adjoint of a linear operator $\hat{A}$, denoted by $A^\dagger$, is an operator that satisfies $$\int_{a}^{b}\psi_1^*(\hat{A}\psi_2)dx =\int_{a}^{b}(\hat{A}^\dagger\psi_1)^*\psi_2 dx+\text{boundary terms}\tag{1}$$ where the boundary terms are evaluated at the end-points of the interval $[a,b]$.

• An operator is said to be self-adjoint if $A^\dagger=A$. Therefore, for self-adjoint operators, $$\int_{a}^{b}\psi_1^*(\hat{A}\psi_2)dx -\int_{a}^{b}(\hat{A}\psi_1)^*\psi_2 dx=\text{boundary terms}.\tag{2}$$

• In addition, if certain boundary conditions are met by the function $\psi_1$ and $\psi_2$ on which the self-adjoint operator acts, or by the operator itself, such that the boundary terms vanish, then the operator said to be hermitian in the interval $a\leq x\leq b$. In that case, $$\int_{a}^{b}\psi_1^*(\hat{A}\psi_2)dx =\int_{a}^{b}(\hat{A}\psi_1)^*\psi_2 dx.\tag{3}$$

My question is that in terms of Dirac's abstract bra and ket notation, how does one write each of these defining equations?