Continuity and Schatten--von Neumann \(p\)-class membership of Hankel operators with anti-holomorphic symbols on (generalized) Fock spaces (Q2492984)

Let \(L^2_m:=L^2({\mathbb C},| z| ^m)\) denote the space of \(g\)-measurable functions with respect to the Lebesgue measure \(d\lambda\) in \(\mathbb C\) such that \[ \| g\| _m^2:=\int_{\mathbb C} | g(z)| ^2 e^{-| z| ^m} d\lambda(z)<\infty, \] and let \(A^2_m:=A^2({\mathbb C},| z| ^m)\) denote its subspace consisting of entire functions. These weighted Bergman spaces are called generalized Fock spaces, the case \(m=2\) being the classical Fock space. The paper deals with Hankel operators \(H_{\overline{f}}:A^2_m\rightarrow {A^2_m}^\perp\), \(H_{\overline{f}} h=({\text{Id}}-P) (\overline{f} h)\), with anti-holomorphic symbol \(\overline{f}=\sum_{k=0}^\infty b_k \overline{z}^k\in L^2_m\) and \(P:L^2_m \overset{\perp}{\rightarrow}A^2_m\) standing for the Bergman projection. The following results are proved: (i) Let \(\overline{f}=\sum_{k=0}^\infty b_k \overline{z}^k \in L^2({\mathbb C},| z| ^m)\) and suppose that \(f\) is not a polynomial. Then the Hankel operator \(H_{\overline{f}}\) is unbounded. (ii) Let \(\overline{f}=\sum_{k=0}^N b_k \overline{z}^k\). Then \(H_{\overline{f}}\) is bounded, if \(2N\leq m\) and \(H_{\overline{f}}\) is unbounded if \(2N>m\). Moreover, \(H_{\overline{f}}\) is compact if \(2N<m\) and \(H_{\overline{f}}\) fails to be compact if \(2N=m\). At the end of the paper, in the case of monomials, the Schatten--von Neumann \(p\)-class membership is characterized. For \(p>0\) and \(2k<m\), the Hankel operator \(H_{\overline{z}^k}\) belongs to the Schatten--von Neumann \(p\)-class if and only if \[ \sum_{n=k}^\infty\left(\frac{c_{n+k}^2}{c_n^2}-\frac{c_n^2}{c_{n-k}^2}\right)^{p/2} <\infty, \] where \(c_n=\| z^n\| _m^2\), \(n\in {\mathbb N}\), are the so-called moments. Some consequences for the case of the classical Fock space (\(m=2\)) are derived and it is shown that this result generalizes the one obtained by the second author [Proc.\ Am.\ Math.\ Soc.\ 132, No.~8, 2399--2409 (2004; Zbl 1068.47032)] for the case \(p=2\).