The essential norm of Hankel operator on the Bergman spaces of strongly pseudoconvex domains
Let \(D\) be a bounded strongly pseudoconvex domain with smooth boundary in \(C^n\), \(n\geq 2\); \(H(D)\) be the space of analytic functions on \(D\), \(A^2(D)= L^2(D)\cap H(D)\) be the Bergman space, \(P\) be the Bergman projection from \(L^2(D)\) onto \(A^2(D)\), \(H_f\) be the Hankel operator with symbol \(f\in L^2(D)\), \[ H_f(g)= fg- P(fg),\;\forall g\in H^\infty(D). \] And \(\|H_f\|_e\) is the distance in the norm operators on \(L^2(D)\), from \(H_f\) to the subspace of compact operators. For each \(f\in L^2(D)\), let \(F_f(z)\) be \[ F_f(z)= \text{inf}\Biggl\{\Biggl({1\over |B(z)|} \int_{B(z)}|f- h|^2 dV\Biggr)^{1/2}; h\in H(D)\Biggr\}, \] where \(B(z)= \{w\in D; d(z, z_0)< r_0\}\) is a Berman metric ball in \(D\). And: \[ BDA= \{f\in L^2(D); F_f(z)\text{ is bounded}\},\;VDA= \{f\in L^2(D); F_f(z)\to_{z\to\partial D} 0\}. \] In this paper, the author proves that: \[ \begin{aligned} & \|H_f\|_e\cong \text{inf}\{\|H_f- H\|; H\text{ compact Hankel operator}\}\\ &\cong \text{inf}\{\|f- g\|_{BDA}; g\in VDA\}\cong \text{inf}\{\|F_{f-g}\|_\infty; g\in VDA\}.\end{aligned} \] This result extends a theorem in the disk by Lin and Rochberg.
scientific article; zbMATH DE number 1484691