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

〈*T**x*, *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.

〈*T**x*, *y*〉 = 〈*x*, *T**y*〉

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 (*T**x*_{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*,

*T**x* = ∑_{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*T**x*=*λ**x*; - (resolvent) the operator
*T*−*λ*has bounded inverse (*T*−*λ*)^{−1}.