fa.functional analysis - Application of bounded spectral theory.

I think Helge's answer cuts to the historical heart of the matter: solution operators for various differential equations tend to be bounded, non-compact operators (obtained in many cases from an unbounded differential operator via the functional calculus), and it is often quite useful from that point of view to know something about their spectra. This is one reason why the theory of elliptic operators over non-compact spaces is more complicated than the corresponding theory for compact spaces: one has to deal with the fact that the eigenvalues of the solution operators can accumulate at 0.

What I find more persuasive, however, are the ways in which spectral theory mediates the relationship between functional analysis and geometry. In many cases you will miss this relationship unless you ignore compact operators entirely. The celebrated Atiyah-Singer Index Theorem provides a particularly dramatic example of this phenomenon, but I'll focus on more digestible examples (A-S is more about spectral theory for unbounded operators anyway).

First, consider the classical Toeplitz operators. Given a complex valued function $g in C(S^1)$, the Toeplitz operator $T_g$ with symbol $g$ is defined as follows. Form the Hardy space $H^2(S^1)$ by considering the $L^2$-closure of the space of polynomial functions on $S^1$, regarding $S^1$ as a subspace of $mathbb{C}$, and let $P$ denote the orthogonal projection from $L^2(S^1)$ to $H^2(S^1)$. Then define $T_g: H^2(S^1) to H^2(S^1)$ to be $T_g(f) = T(fg)$. This is a bounded operator, and the classical Toeplitz index theorem asserts that its Fredholm index (the dimension of its kernel minus the dimension of its cokernel) is precisely the winding number of $g$. Thus an analytic invariant of the Toeplitz operator with symbol $g$ calculates a topological invariant of $g$.

That result by itself isn't heavy on the spectral theory. The connection with spectral theory is revealed by a more refined statement. Recall that the essential spectrum of a bounded operator $T$ on a Hilbert space $H$ is the spectrum of the image of $T$ in the Calkin algebra $Q(H)$ (which is the space of bounded operators on $H$ modulo the space of compact operators). A consequence of the Toeplitz index theorem (and its proof) is the fact that the essential spectrum of the Toeplitz operator $T_g$ is precisely the range of $g$. While this statement alone is scant evidence, this suggests a deep relationship between the essential spectrum of a bounded operator and geometry. This line of thinking culminates in the Brown-Douglas-Fillmore theorem, which makes the following startling assertion. Let $X$ be a nonempty subset of the complex plane, and define $Ext(X)$ to be the space of essential unitary equivalence classes of essentially normal operators with essential spectrum $X$ (here "essential" always means "modulo compact operators"). Direct sum of operators gives $Ext(X)$ the structure of a commutative semigroup, and the BDF theorem asserts that $Ext(X)$ is naturally isomorphic to the space $Hom(pi^1(X), mathbb{Z})$ of group homomorphisms between the first cohomotopy group of $X$ and $mathbb{Z}$. (Note: it is not even obvious that $Ext(X)$ has a zero element!) Thus spectral theory helps to classify certain kinds of bounded operators mod compacts in a particularly beautiful way (via algebraic topology).

As mentioned above, there are also fruitful interactions between functional analysis and geometry - mediated by spectral theory - which flow from analysis to geometry. Aside from Atiyah-Singer, there are fruitful generalizations of the Toeplitz index theorem along these lines. But let me give a different sort of example in the theory of hyperbolic diffeomorphisms.

Informally, a diffeomorphism $f: M to M$ on a smooth manifold $M$ is said to be Anosov (or uniformly hyperbolic) if $M$ admits transverse stable and unstable foliations for the action of $f$. Prototypical examples of Anosov diffeomorphisms on the 2-torus can be obtained by considering $2 times 2$ matrices with integer entries and irrational eigenvalues. It turns out that spectral theory has a great deal to say about smooth dynamical systems in general and Anosov diffeomorphisms in particular.

Given any diffeomorphism $f: M to M$, consider the bounded operator $f_*$ on the Banach space $Gamma^0(TM)$ of continuous vector fields on $M$ defined by

$(f_*v)(x) = df(v)(f^{-1}(x))$

If the non-periodic orbits of $f$ are dense in $M$, then a theorem of Mather asserts that the spectrum of $f_*$ is a disjoint union of finitely many annuli centered at the origin. If $H_i$ is the invariant subspace for $f_*$ corresponding to the $i$th annulus then the subspaces $E_i(x)$ of $T_x M$ consisting of the vectors $v(x)$ for $v in H_i$ form a $df$ invariant continuous distribution on $M$, and the direct sum of the $E_i(x)$'s gives the whole tangent space $T_x M$. So the Mather spectral theory of $f$ is very closely related to its dynamics. Indeed, one can characterize the Anosov diffeomorphisms as precisely those $f$ for which $1$ is not in the spectrum of $f_*$. Pesin used this idea to prove that Anosov diffeomorphisms are structurally stable, meaning they form an open subset of the full diffeomorphism group of $M$ (so that a small perturbation of an Anosov diffeomorphism is still Anosov). The same strategy also works for partially hyperbolic dynamical systems, which have a slightly different spectral characterization.