Compact Hankel operators with conjugate analytic symbols

Abstract

In 1986 S. Axler [3] proved that forf∈L ²_a the Hankel operator$H_{\bar f} :L_a^2 \to (L^2 )^ \bot$ is compact if and only iff is in the little Bloch space {itB}{in0}. In this note we show that the same is true for$H_{\bar f} :L_a^p \to L^p$, 1<p<∞. Moreover we prove that$H_{\bar f} :L_a^1 \to L^1$ is ⋆-compact if and only if$|f'(z)|(1 - |z|^2 )\log \tfrac{1}{{1 - |z|^2 }} \to 0$ as |z|→1⁻.