It is with some sort of reverential fear that I've come here to write. I've been reading you for a long time, but writing is another story... In any case, I suppose it is too late now to back out!

Then, I am looking for (as many as possible) references to known "different" proofs of the classical spectral theorem for compact operators with a special focus on the point where we are given to show that all non-zero elements in the spectrum are, in fact, eigenvalues. I am well aware of the "standard one" (as basically phrased in this Wikipedia entry) and I have tidings of a proof based on the Fredholm alternative (though I don't know any explicit reference in this case). Indeed, I'm wondering if there are some others around. Thanks so much for any clues.

It is with some sort of reverential fear that I've come here to write. I've been reading you for a long time, but writing is another story... In any case, I suppose it is too late now to back out!

Then, I am looking for (as many as possible) references to known "different" proofs of the classical spectral theorem for compact (linear) operators (on complex Banach spaces) with a special focus on the point where we are given to show that all non zero elements in the spectrum are, in fact, eigenvalues. I am well aware of the "usual one" (as basically drafted in this Wikipedia entry - just look at the ideas since at present the proof is flawed in some parts, as outlined by Prof. Johnson below in the comments) and I have tidings of a proof based on the Fredholm alternative (though I don't know any explicit reference in this case). Indeed, I'm wondering if there are some others around. Thanks so much for any clues.