The Spectral Theorem and the Fredholm Alternative

My recent reading on uncertainty principles in quantum mechanics, where a lot of things are expressed in terms of operators on Hilbert spaces, has required me to refresh my memory with the basic facts about the spectral theory of self-adjoint and compact operators on Hilbert and Banach spaces.

Let H and K be Hilbert spaces and T: H → K a linear operator. The adjoint of T is the linear operator T^∗: K → H defined by

〈Tx, y〉 = 〈x, T^∗y〉

for all x ∈ H and y ∈ K. An operator T: H → H is said to be self-adjoint if T = T^∗, i.e.

〈Tx, y〉 = 〈x, Ty〉

for all x, y ∈ H. If we fix bases of H and K, then the matrix of T^∗ is the conjugate transpose of that of T. Self-adjoint operators have the following very nice eigenvalue/eigenvector property:

Theorem. If T: H → H is a self-adjoint operator on a Hilbert space H, then all the eigenvalues of T are real and the eigenvectors corresponding to distinct eigenvalues are orthogonal in H.

Another property that plays very nicely with self-adjointness is compactness:

Let X and Y be Banach spaces. A linear operator T: X → Y is said to be a compact operator if, for every bounded set B ⊆ X, the image T(B) ⊆ Y is a compact subset of Y. Equivalently, every norm-bounded sequence (x_n)_n∈ℕ in X is such that the image sequence (Tx_n)_n∈ℕ in Y has a convergent subsequence.

Clearly, compact operators are always bounded. Operators that are both compact and self-adjoint are especially nice:

Theorem. If T is a compact self-adjoint operator on a Hilbert space, then either ‖T‖ or −‖T‖ is an eigenvalue of T. Furthermore, the set of all eigenvalues of T (the point spectrum of T) is either finite or consists of a countable sequence tending to 0.

[Incidentally, while I had no problem remembering the “furthermore” part about the point spectrum, I had completely forgotten the simpler fact that, up to sign, the operator norm of T is always one of its eigenvalues.]

The next result, the Spectral Theorem, is essentially the “diagonalization” result for compact, self-adjoint operators, analogous to results from finite-dimensional linear algebra on the diagonalization of (conjugate-)symmetric square matrices.

Spectral Theorem. If T is a compact self-adjoint operator on a Hilbert space H, then there exists a finite or infinite orthonormal sequence (x_n) of eigenvectors of T with corresponding real eigenvalues (λ_n) such that, for all x ∈ H,

Tx = ∑_n λ_n 〈x, x_n〉 x_n.

Note that the orthonormal sequence of the Spectral Theorem can be extended by the usual basis extension and orthogonalization procedures to an orthonormal basis of H.

The other key fact about eigenvalue problems for compact (not necessarily self-adjoint) operators is the Fredholm Alternative, which in finite-dimensional settings follows from the Rank-Nullity Theorem. The Fredholm Alternative asserts a dichotomy: either λ is an eigenvalue of T, or else it lies in the domain of the resolvent of T. More precisely,

Fredholm Alternative. Let T: X → X be a compact operator on a Banach space X, and let λ ∈ ℂ be non-zero. Then precisely one of the following holds true:

• (eigenvalue) there is a non-zero solution x ∈ X to the eigenvalue equation Tx = λx;
• (resolvent) the operator T−λ has bounded inverse (T−λ)⁻¹.