Finite rank little Hankel operators on Bergman-type spaces
For a domain \(\Omega\) in \(\mathbb C^n\) and \(\phi\in L^\infty(\Omega)\), the big Hankel operator \(H_\phi\) and the little Hankel operator \(S_\phi\) on the Bergman space \(L^2_a(\Omega)\) of all square-integrable holomorphic functions on \(\Omega\) are defined by \(H_\phi f=(I-P)(\phi f)\) and \(S_\phi f=P(J(\phi f))\), respectively, where \(P:L^2(\Omega)\to L^2_a(\Omega)\) is the orthogonal projection and \(Jf(z)=f(\overline z)\). In~the present paper, the author establishes a number of facts about these operators, their kernels and their being of finite rank. In~particular, the operators \(S_\phi\) are characterized as the solutions of the equation \(ST_z=T^*_zS\), where \(T_z\) is the operator of multiplication by~\(z\); the kernel of \(H_\phi\) is shown to be trivial if \(H_\phi\neq0\); and \(S_\phi\) is of finite rank only if \(P\overline \phi\) is a linear combination of the Bergman kernels at some points and of their derivatives. Reviewer's~remark: The~reader should be careful about a number of minor inaccuracies in the paper --- for instance, in~Theorem~3.2 the subspace \(\mathcal M\) must be assumed to be nontrivial; in Proposition~3.6, one must count multiplicities of the zeroes in \(\mathcal Z(\mathcal I)\); and, finally, it should be made clearer what domains \(\Omega\) the various assertions apply~to: the Introduction suggests that these are all bounded symmetric domains in \(\mathbb C^n\) as well as all simply connected Jordan domains in~\(\mathbb C\), but, e.g.,~Theorem~2.3 does not quite make sense in dimensions greater than~1, while some of the properties of the inner functions referred to in Theorems~3.7--3.10 have in fact even been established only for the unit disc.
scientific article; zbMATH DE number 5009702