The Adamyan–Arov–Krein theorem: Vectorial variant

S. R. Treil

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XIV, Tome 141 (1985), pp. 56-71

Résumé

One obtains the following description of the $s$–numbers of the vectorial Hankel operators $H_{\varphi}$, $\varphi\in L^{\infty}(E_1,E_2)$. Theorem 1. {\it $s_n(H_{\varphi})=\inf\{\|H_{\varphi}-H_{\psi}\|:\operatorname{rank} H_{\psi}\le n\}$}. The theorem generalizes the known Adamyan–Arov–Krein result and in the case $\min(\dim E_1,\dim E_2)<\infty$ has been proved by Ball and Helton. One obtains a constructive description of the Hankel operators of finite rank and one gives a formula for the rank of such an operator.