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Schatten class Hankel operators on the Bergman spaces of strongly pseudoconvex domains. (English) Zbl 0802.47022
Consider on a strongly pseudoconvex domain in \(\mathbb{C}^ n\) the Hankel operator \(H_{\overline{f}}\) given by \(H_{\overline {f}} (g)= (I-P) (\overline {f} g)\), where \(f\) is a square-integrable holomorphic function, and \(P\) is the Bergman projection. The author shows that if \(p>2n\), then \(H_{\overline {f}}\) belongs to the Schatten class \(S_ p\) if and only if \(f\) belongs to the holomorphic Besov space \(B_ p\), while if \(p\leq 2n\), then \(H_{\overline {f}}\in S_ p\) if and only if \(f\) is a constant function.
The result was obtained independently by Marco M. Peloso [Ill. J. Math. 38, No. 2, 223-249 (1994)].
The proof uses integral formulas for solving the \(\overline {\partial}\)- equation.
For the special case of the unit ball, the result was found previously by R. Wallsten [Ark. Mat. 28, No. 1, 183-192 (1990; Zbl 0705.47023)], J. Arazy, S. Fisher, S. Janson, and J. Peetre [J. Lond. Math. Soc., II. Ser. 43, No. 3, 485–508 (1991; Zbl 0747.47019)], and K. Zhu [Am. J. Math. 113, No. 1, 147-167 (1991; Zbl 0734.47017)].

MSC:
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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