3
$\begingroup$

In the spectral theorem for compact self-adjoint linear operators $T:H\to H$ (as stated in Conway's book), the Hilbert space $H$ can be real or complex.

However, in the spectral theorem for bounded self-adjoint linear operators $T:H\to H$ (as stated in Bachman's book), the Hilbert space $H$ have to be complex.

Is this assumption really needed? Is there any version for bounded self-adjoint linear operators defined on real spaces?

$\endgroup$
  • $\begingroup$ I guess for a real Hilbert space H, you can always complexify it to get a complex Hilbert space K and extend the your operator to be defined on K. After this extension, your operator should still be self-adjoint, then you can apply the spectral theory. $\endgroup$ – lye012 May 12 at 21:42
  • $\begingroup$ math.stackexchange.com/questions/638216/… $\endgroup$ – DisintegratingByParts May 13 at 3:32

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.