Weyl's theorem on compact perturbations

Remarks

For a self-adjoint operator, $\lambda$ lies in the essential spectrum if and only if there is a Weyl sequence: an orthonormal sequence $(x_n)$ in $H$ with $\norm{(A - \lambda) x_n} \to 0$. An orthonormal sequence tends to $0$ weakly, and a compact operator maps weakly null sequences to norm null sequences, so $\norm{K x_n} \to 0$. Hence a Weyl sequence for $A$ at $\lambda$ is also one for $A + K$, and the symmetric argument gives the reverse inclusion. The theorem also holds for unbounded self-adjoint $A$, with $A + K$ defined on the domain of $A$. In other words, a compact perturbation of a self-adjoint operator can only change its discrete spectrum, the isolated eigenvalues of finite multiplicity. With the definition of the essential spectrum through Fredholm operators, the equality holds for every bounded operator $A$ on a Banach space and every compact $K$. Indeed, $A + K - \lambda I$ is Fredholm exactly when $A - \lambda I$ is. Example: on $\ell^2$, the zero operator has essential spectrum $\set{0}$, and so does every compact self-adjoint operator, such as the diagonal operator with entries $1/n$. Without compactness the conclusion fails: a self-adjoint projection $P$ of infinite rank and infinite corank has $\sigma_{\mathrm{ess}}(0 + P) = \set{0, 1}$, while $\sigma_{\mathrm{ess}}(0) = \set{0}$.

Going deeper

For a student

The theorem is a basic tool of mathematical physics. For a Schrödinger operator $-\Delta + V$ with a potential $V$ that decays at infinity, one shows that $V$ is a relatively compact perturbation of $-\Delta$. A version of Weyl's theorem for relatively compact perturbations then gives $\sigma_{\mathrm{ess}}(-\Delta + V) = [0, \infty)$, so the negative spectrum consists of isolated eigenvalues, the bound states.