I was looking around different textbooks and websites for the definition of a Hermitian adjoint. All the resources that I have checked including the one I am studying at the moment (Jeevanjee's Intro. to Tensors and Group Theory for Physicists pg. 48 and footnote on pg. 120) assume a positive-definite non-degenerate Hermitian form to define a Hermitian adjoint. I was wondering if you can at all define a Hermitian adjoint when the Hermitian form is not positive definite, and if there is a problem with that what is it?
Thank you
As you have noticed, there's no problem with defining adjoint operators for non-degenerate Hermitian forms.
The reason people usually only consider positive-definite Hermitian forms has more to do with analysis. If we have a positive-definite form, it defines the standard Euclidean topology on the vector space. We also get the nice property that $\|x\| = 0$ implies that $x = 0$, and the Cauchy-Schwarz inequality. All those are very handy when doing any kind of analysis.
I suppose the reason most people then explicitly use a positive-definite form when defining adjoint operators is that talking about a general non-degenerate form would be an aside that has little to do with what they want to focus on.