Hankel operator

Summary

A Hankel operator is a bounded operator on $\ell^2$ whose matrix $(a_{j+k})_{j,k \ge 0}$ is constant along each anti-diagonal, an infinite Hankel matrix. Up to a unitary identification of the spaces, the same operators arise in a function form. They are the maps $f \mapsto (I - P)(\varphi f)$ from the Hardy space $H^2$ to $L^2 \ominus H^2$ on the circle. Here $\varphi$ is a bounded symbol and $P$ is the projection onto $H^2$. Hankel operators are close relatives of Toeplitz operators and appear in rational approximation, control theory and prediction theory. Nehari's theorem describes the bounded ones, and Kronecker's theorem the ones of finite rank.