Compact Hankel operators with bounded symbols

Hagger, R. and Virtanen, J. (2021) Compact Hankel operators with bounded symbols. Journal of Operator Theory, 86 (2). pp. 317-329. ISSN 0379-4024 doi: 10.7900/jot.2020apr27.2276

Abstract/Summary

We discuss the compactness of Hankel operators on Hardy, Bergman and Fock spaces with focus on the differences between the three cases, and complete the theory of compact Hankel operators with bounded symbols on the latter two spaces with standard weights. In particular, we give a new proof (using limit operator techniques) of the result that the Hankel operator Hf is compact on Fock spaces if and only if Hf¯ is compact. Our proof fully explains that this striking result is caused by the lack of nonconstant bounded analytic functions in the complex plane (unlike in the other two spaces) and extends the result from the Fock-Hilbert space to all Fock-Banach spaces. As in Hardy spaces, we also show that the compactness of Hankel operators is independent of the underlying space in the other two cases.