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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