---
tag: MJ48S56VQ7
kind: definition
name: Hankel operator
msc: [47B35]
defines: P4EKQS312D
uses: [PQ1D8FQB3E, 08P6T2WH6R, 03TM135G0N]
sources:
  - url: https://en.wikipedia.org/wiki/Hankel_matrix
---

## Definition

Let $a = (a_n)_{n \ge 0}$ be a sequence of complex numbers.
Let $\ell^2 = \ell^2(\set{0, 1, 2, \ldots})$ be the [[space of square-summable sequences|08P6T2WH6R]] with standard basis $(e_k)_{k \ge 0}$.
The *Hankel operator* $\Gamma_a \colon \ell^2 \to \ell^2$ is the [[bounded linear operator|03TM135G0N]] with the [[Hankel matrix|PQ1D8FQB3E]] $(a_{j+k})_{j, k \ge 0}$, that is,
$$\ip{\Gamma_a e_k}{e_j} = a_{j+k} \quad \text{for all } j, k \ge 0,$$
when such a bounded operator exists.

## 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 $H^2$.