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Schatten class of Hankel and Toeplitz operators on the Bergman space of strongly pseudoconvex domains. (English) Zbl 0833.47017
Curto, Raúl E. (ed.) et al., Multivariable operator theory. A joint summer research conference on multivariable operator theory, July 10-18, 1993, University of Washington, Seattle, WA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 185, 237-257 (1995).
Let \(D\) be a strongly pseudoconvex domain in \(\mathbb{C}^n\) with the strictly plurisubharmonic defined function \(\rho\). Let \(dV\) denote the Lebesgue volume on \(D\). The Bergman space on \(D\) is the space \(A^2= L^2 (D, dV)\cap {\mathcal H}\), where \({\mathcal H}\) is a collection of holomorphic functions on \(D\). For a function \(f\in L^1 (dV)\) the Hankel operator \(H_f\) is densely defined on \(A^2\) by the formula \[ H_f (g) (z)= (I- P) (fg)= f(z)- \int_D K(z,w) f(w) g(w) dV(w), \] where \(K(z, w)\) is the Bergman kernel for \(D\) and \(P\) is the Bergman projection. Let \(T_\mu\) be the Toeplitz operator defined by \(\langle T_\mu f,g \rangle= \int_D f\overline {g} d\mu\) where \(\mu\) is the measure \(|f|^2 dV\).
The authors give conditions when Toeplitz and Hankel operators belong to the Schatten classes \({\mathcal S}_p (A^2)\), \(0< p<\infty\).
For the entire collection see [Zbl 0819.00022].

MSC:
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
46E20 Hilbert spaces of continuous, differentiable or analytic functions
32T99 Pseudoconvex domains
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