Hankel operator

Remarks

Example: the Hilbert matrix, with $a_n = 1/(n + 1)$, defines a bounded Hankel operator; Hilbert's inequality gives $\norm{\Gamma_a} = \pi$.

Non-example: the identity matrix is not a Hankel matrix. Its entries at $(1, 1)$ and $(0, 2)$ have the same index sum $2$, but they are $1$ and $0$.

Nehari's theorem: $\Gamma_a$ is bounded if and only if there is an essentially bounded function $\psi$ on the unit circle with $\hat\psi(n) = a_n$ for all $n \ge 0$. Then $\norm{\Gamma_a}$ is the least $\norm{\psi}_\infty$ over such $\psi$, where $\hat\psi(n)$ is the $n$-th Fourier coefficient. Kronecker's theorem: $\Gamma_a$ has finite rank if and only if the power series $\sum_{n \ge 0} a_n z^n$ is a rational function. In the function form, for a bounded symbol $\varphi$ on the circle, the Hankel operator is $H_\varphi f = (I - P)(\varphi f)$ from $H^2$ to $L^2 \ominus H^2$, with $P$ the orthogonal projection onto $H^2$.