What is compact operator?

Example 15.3

(a) Diagonal operators on $\ell^2$: $T(x_n) = (\lambda_nx_n)$ is compact iff $\lambda_n \to 0$ (Exercise 15.2). (b) Kernel operators on $\mathcal C(\intcc01)$: compact by Ascoli (Exercise 7.7). (c) Hilbert–Schmidt operators: for $k \in L^2(\intcc01^2)$,

defines a compact operator on $L^2(\intcc01)$ with $\vertiii{T_k} \leq \norm k_{L^2}$ (Exercise 15.4: truncating the basis expansion of $k$ exhibits $T_k$ as a limit of finite-rank operators).

Example 15.9

On $L^2(\intcc01)$, let $Tf(x) = \int_0^1\min(x, y)f(y)\dd y$: a Hilbert–Schmidt operator with real symmetric kernel: compact and self-adjoint. Solving $Tf = \lambda f$: the relation $\bigl(Tf\bigr)(x) = \int_0^xyf(y)\dd y + x\int_x^1f(y)\dd y$ shows $u = Tf$ satisfies $u'' = -f$ (two differentiations, legitimate for continuous $f$, and $Tf$ is continuous for $f \in L^2$: dominated convergence), with $u(0) = 0$ and $u'(1) = 0$. So eigenfunctions solve $\lambda u'' = -u$, $u(0) = 0$, $u'(1) = 0$:

and the spectral theorem asserts — with no Fourier theory — that these sines form an orthonormal basis of $L^2(\intcc01)$ after normalization (the kernel of $T$ is $0$: $Tf = 0$ forces, by the two differentiations, $f = 0$ a.e.). The weekend problem runs the same circle of ideas for the vibrating string and extracts $\zeta(2)$ from the trace.