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.