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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

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  • $\begingroup$ Have you tried defining that the same way as for the positive-definite case and seeing if that works? $\endgroup$ – Gunnar Þór Magnússon Jan 31 at 6:34
  • $\begingroup$ Yes, I have but I can't find any problems or inconsistencies with assuming an existence of a hermitian adjoint on a non-positive-definite nondegenerate Hermitian Form. $\endgroup$ – jzme1234 Feb 6 at 1:58
  • $\begingroup$ Congratulations on your success. 🎉 $\endgroup$ – Gunnar Þór Magnússon Feb 6 at 7:56
  • $\begingroup$ But these books only define it for positive definite hermitian forms, i was wondering why they do that... $\endgroup$ – jzme1234 Feb 14 at 19:53
  • $\begingroup$ Positive-definite (or negative) forms have the property that $\|x\| = 0$ means $x = 0$, which is very nice for some arguments. $\endgroup$ – Gunnar Þór Magnússon Feb 15 at 5:26
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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.

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