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Hankel operators on weighted Bergman spaces on strongly pseudoconvex domains. (English) Zbl 0812.47023
The paper concerns the study of the Hankel operator \(H_ f\) and the non- orthogonal Hankel operator \(\widetilde H_ f\); i.e. \[ H_ f g(z)= (I- P)(\widetilde f g)(z);\;\widetilde H_ f g(z)= (I- \widetilde P)(\widetilde f g)(z), \] where \(P(\widetilde P)\) is the (non-)orthogonal projection \(L^ 2\nu\to A^{2,\nu}\) (\(A^{2,\nu}\) the weighted Bergman space = the closed subspace in \(L^ 2_ \nu\) of holomorphic functions; \(L^ 2_ \nu= L^ 2\)-space \(L^ 2(\Omega,dm_ \nu)\) on a \(C^ \infty\)-bounded strongly pseudoconvex domain \(\Omega\subset \mathbb{C}^ n\) and \(dm_ \nu= |\rho(z)|^ \nu dm\), \(dm=\) Lebesgue volume form).
The main results of the paper are
Theorem 1. Let \(f\in {\mathcal H}(\Omega)\). Then the following are equivalent.
i) \(f\in {\mathcal B}\);
ii) \(H_ f\) is bounded;
iii) \(\widetilde H_ f\) is bounded.
Theorem 2. Let \(f\in {\mathcal H} (\Omega)\). Then the following are equivalent.
i) \(f\in {\mathcal B}_ 0\);
ii) \(H_ f\) is compact;
iii) \(\widetilde H_ f\) is compact.
(Here \({\mathcal H}(\Omega)=\) holomorphic functions space on \(\Omega\); \[ {\mathcal B}= \{f| f\in {\mathcal H}(\Omega),\;\sup_{z\in \Omega} | \widetilde D f(z)|< \infty\} \] with \(\widetilde D\) the covariant derivative, and \(f\in {\mathcal B}_ 0\) if \(f\in {\mathcal B}\) with \(\lim_{z\in \partial\Omega} |\widetilde Df(z)|= 0\)).
Reviewer: R.Salvi (Milano)

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)