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I would like to discuss the “next entry” in the parallel texts that Masoud was presenting in his post. On the function theory side we are talking about “real and complex variables”. A perfect book to get introduced to that is “real and complex analysis” by W. Rudin (McGraw-Hill). It is a classic and remains one of the best entrance doors to the subject. What one learns is the constant interplay between the “real variable” techniques such as the Lebesgue integral, differentiability almost everywhere, etc.. and the “complex variable” techniques. There is a saying of André Weil like “The complex world is beautiful, the real world is dirty”. One might then be tempted to ignore the “real world” and only work in the complex variable set-up where “any” function is holomorphic and hence infinitely differentiable etc… That’s fine, and one can go some distance with that, except that most of the deep results in complex analysis do rely on real analysis.
Now what about the next entry in the parallel text? It is
Complex variable…………….Operator on Hilbert space
Real variable……………………..Self-adjoint operator
where I have slightly rewritten the previous entry
functions f: X -> C ……………..operators on Hilbert space
of Masoud’s post to stress that the right column gives an ideal model for what the loose notion of a “variable” is… The set of values of the variable is the spectrum of the operator, and the number of times a value is reached is the spectral multiplicity. Continuous variables (operators with continuous spectrum) coexist happily with discrete variables precisely because of non-commutativity of operators.
The holomorphic functional calculus gives a meaning to f(T) for all holomorphic functions f on the spectrum of T, and a deep result controls the spectrum of f(T). The really amazing fact is that while for general operators T in Hilbert space the only functions f(z) that can be applied to T are the holomorphic ones (on the spectrum of T), the situation changes drastically when one deals with self-adjoint operators: for T=T* the operator f(T) makes sense for any function f! You can take a pencil and draw the graph of a function, it does not need to be continuous…nor even piecewise continuous, just anything you can name will do….(at the technical level the only requirement on f is that it is universally measurable but nobody can construct explicitly a function which does not fulfill this condition!)…Moreover a bounded operator is a function of T (ie is of the form f(T) ) if and only if it shares all the symmetries of T (ie if it commutes with all operators that commute with T ).
I remember that, at a very early stage of my encounter with mathematics, it is this very fact that convinced me of the power of the Hilbert space techniques in close relation with the adjoint operation T -> T*. This was enough to resist the temptation of starting directly in the “complex world” of algebraic geometry which was attracting most beginners at that time, following the aura of Grothendieck, who described so well his first encounter with that world:
“Je me rappelle encore de cette impression saisissante (toute subjective certes), comme si je quittais des steppes arides et revèches, pour me retrouver soudain dans une sorte de “pays promis” aux richesses luxuriantes, se multipliant à l’infini partout où il plait à la main de se poser, pour cueillir ou pour fouiller….”