The Forward Direction of "V: a + iTa=b+iTb, iff a=b"

[CoachZ]

Let V be a complex inner product space and T a self-adjoint linear operator on V.

I'm trying to show a + iTa = b + iTb, iff a = b. The converse is trivial. The forward direction is getting me for some reason. Perhaps it's too late on a Friday night that my mind is completely gone. Any suggestions...

 

[HallsofIvy]

Let V be a complex inner product space and T a self-adjoint linear operator on V.

I'm trying to show a + iTa = b + iTb, iff a = b. The converse is trivial. The forward direction is getting me for some reason. Perhaps it's too late on a Friday night that my mind is completely gone. Any suggestions...

So a- b= -iT(a- b) and T(a- b)= i(a- b). Either a- b= 0 or i is an eigenvalue of T. But all eigenvalues of self-adjoint operators are real.

 

[CoachZ]

Haha, I woke up this morning thinking about I + iT and I - iT, and how they are both non-singular, which uses the theorem of real eigenvalues, and I was thinking that a + iTa = b + iTb probably uses similar concepts and ideas to solve. Thanks for the help!