As mentioned in the previous answer, the Atiyah-Singer index theorem is an excellent answer to your question. I would like to convince you that, in a sense, it is probably the only answer to your question. Fortunately that one theorem admits so many applications, generalizations, and elaborations that it almost becomes an area of mathematics unto itself (particularly when infused with the tools of C*-algebra theory).
My first remark is that K-theory is an inherently global tool - its power lies in the fact that it is built from but insensitive to the details of local geometry. From what I understand about PDE theory many of the interesting questions live on open balls in Euclidean space, about which algebraic topology in general has little new to contribute. Even when one considers boundary value problems where the geometry gets a little more interesting, the challenges are usually local on the boundary (i.e. the concern is with smoothness, not interesting global structure).
Once we have accepted that we are looking for a global answer to your question, it is natural to ask: is there a sense in which PDE's organize themselves into a full-fledged (co)homology theory? This, after all, is the way that topology usually interacts with other parts of mathematics: one begins with objects whose structure one wants to globalize (e.g. embedded loops, differential forms, vector bundles...) and one aims to build algebraic invariants out of those objects. In the case of PDE's the answer is K-homology, a generalized homology theory for the category of manifolds with the property that every first order linear elliptic operator $D$ on a manifold $M$ gives rise to a class $[D]$ in $K_*(M)$. K-homology, as the name suggests, is the homology theory which is naturally dual to K-theory, regarded as a (generalized) cohomology theory on the category of manifolds.
So the question is: what can we do with K-homology? The answer is that one can do quite a lot, but as with many constructions in algebraic topology many of the most interesting results involve pairings between homology and cohomology. The most fundamental pairing between K-homology and K-theory is the so-called index pairing, which takes an elliptic operator and a vector bundle and spits out the Fredholm index of the operator "twisted" by the bundle. The Atiyah-Singer index theorem is really a theorem about the topological properties of this pairing, and consequently it plays a central role in applications of K-homology.
I don't know how useful this will be, but I have some lecture notes that motivate the last three things on your list by first reinterpreting the finite dimensional spectral theorem in terms of the functional calculus. (There is also a section on the spectral theorem for compact operators, but this is just pulled from Zimmer's Essential Results of Functional Analysis.) I gave these lectures at the end of an undergraduate course on functional analysis, though, so they assume familiarity with Banach and Hilbert spaces.
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Suppose you want to find a number $r$ whose square $r^{2}$ is $2$. That has no meaning for numerical analysis because all numbers on a computer are rational, and $\sqrt{2}$ is not rational. It wasn't until the late 1800's that Mathematicians found a logically consistent way to define a real number. But once such a beast could be defined, then one can prove that various algorithms will get you closer and closer to $r$ to $\sqrt{2}$, knowing that it has something to converge to. The existence of such a thing in the extended "real" number system became important to the discussion.
Sobolev spaces are to the ordinary differentiable functions what the real numbers are to the rational numbers. In the late 1800's it was discovered that Calculus of variations didn't have minimizing or maximizing functions. It was the same type of problem: a larger class of functions had to be considered, and the corresponding definitions of integrals had to be extended in order to make sense of and to find a unique minimizer or maximizer that would solve variational problems. So new functions spaces emerged, Lebesgue integration extended the integral expressions to new function classes, and solutions could be found. Once minimizing or maximizing functions could be found, their properties could be deduced, and it validated various algorithms used to find solutions that couldn't converge before because there was nothing to converge to.