In Professor Axler's Linear Algebra Done Right, he separates the proof of the spectral theorem for normal operators on a complex v.s. from the spectral theorem for self-adjoint operators over a real v.s., and I am wondering if each proof also works for the other case.
To my understanding, sketches of the proofs are as follows:
-
Normal operator $T$ on a complex v.s.:
$T$ has an eigenvalue $\lambda$. Schur's theorem says we can express $T$ as an upper triangular matrix (apply the inductive hypothesis of Schur's theorem to the range of $T – \lambda I$ and complete the basis). Result that $||Tx|| = ||T^*x||$ for normal operators gives the result. -
Self-adjoint operator $T$ on a real v.s.:
$T$ has an eigenvalue $\lambda$. Apply the inductive hypothesis of the spectral theorem to the orthogonal complement of the span of an eigenvector (allowed to do this because this is a $T$-invariant subspace) to get result.
It seems to me like the key result is showing the existence of eigenvalues, and each proof would work for the other case (to apply (2) to normal operators, we need to verify that the complement of an eigenspace is $T$-invariant if $T$ is normal). I just want to make sure I'm not missing some subtlety that explains why Professor Axler separated these 2 cases.
Thank you!
Best Answer
Although the proofs of the finite-dimensional complex spectral theorem and the finite-dimensional real spectral theorem can be forced into a unified proof, it seems to me that there are enough distinctive features so that separating the two cases leads to better understanding.
For example, the result that if $T$ in normal then $\|Tx\| = \|T^*x\|$, which is crucially used in the proof of the complex spectral theorem, plays no role in the proof of the real spectral theorem.
As another example, the existence of an eigenvalue for a normal operator on a complex vector space has no connection with the operator being normal, while the existence of a real eigenvalue for a self-adjoint operator on a real vector space is a crucial (and nontrivial) part of many proofs of the real spectral theorem.
For what seem to me to be optimal proofs of the real spectral theorem and the complex spectral theorem, see 7.29 and 7.31 in the new fourth edition of my book Linear Algebra Done Right, which is legally available for free at https://linear.axler.net/.