In my Functional Analysis class, we have been studying self-adjoint compact operators for the past week or so (more specifically their spectrums). I have a geometric idea of what it means for an operator to be compact (the image of any subset is relatively compact in the codomain) and I have a vague geometric notion of the adjoint in my head (define a new operator whose image is the complement of the original operator's kernel and vice versa). I'd like some intuition for what exactly it means for an operator in a Hilbert space to be self-adjoint.
Obviously this isn't a question with a right answer; I'd just like to know how people think about/visualize self-adjointness, and I'd appreciate the thoughts of anyone who's tried to do so!
The notion of an adjoint came out of Lagrange's study of differential equation: https://en.wikipedia.org/wiki/Lagrange%27s_identity_(boundary_value_problem) Techniques of Lagrange arose in the context of integration by parts. $$ \int (Lf)g dx = \int f(L^\dagger g)dx+... $$ Lagrange's techniques were created to study ordinary differential equations. I believe that reduction of order also came out of this study.
When Sturm and Liouville were studying the ODEs arising out of Fourier's separation of variables for the Heat Equation, they found the work of Lagrange on his 'adjoint' or 'adjunct' equation to be particularly helpful in studying the 'symmetric' ODEs arising out of Fourier's separation of variables.
Theorem [Lagrange Adjoint Identity] Let $p$ be continuously differentiable on $(a,b)$ and let $q$ be continuous on $(a,b)$. For any twice continuously differentiable function $f$ on $(a,b)$ define $$ Lf = -\frac{d}{dx}\left(p(x)\frac{df}{dx}\right)+q(x)f. $$ If $f$, $g$ are twice continuosly differentiable on $(a,b)$, then $$ (Lf)g - f(Lg) = \frac{d}{dx}\left\{ p \left(fg'-f'g\right)\right\}. $$
This provided a type of reduction of order for 'symmetric' differential operators $L$.
Sturm and Liouville applied this work to the 'separation of variables' technique developed by Fourier for studying his Heat Equation. They were able to justify a general technique for finding eigenvalues and eigenfunctions of symmetric second order ODEs arising out of Fourier's separation of variables, and they were able to justify the corresponding eigenfunction expansions.
By imposing separated endpoint conditions on an interval $[a,b]$, they found a discrete set of eigenvalues and eigenfunctions arising out of Fourier's equations, and they were able to justify eigenfunction expansions in these solutions. This was a remarkable achievement for the early 1800's, especially considering that the abstract notion of inner product, orthogonality, linearity, and eigenfunction came out of this study and the work of Fourier. All of this came even before the study of symmetric matrices (in fact the theory of symmetric matrices grew out of this work.)