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Let \Omega be a bounded pseudoconvex domain in \mathbb {C}^n with Lipschitz boundary and \phi be a continuous function on \overline {\Omega }. We show that the Toeplitz operator T_{\phi } with symbol \phi is compact on the weighted Bergman space if and only if \phi vanishes on the boundary of \Omega. We also show that compactness of the Toeplitz operator T^{p,q}_{\phi } on \overline {\partial }-closed (p,q)-forms for 0\leq p\leq n and q\geq 1 is equivalent to \phi =0 on \Omega. [ABSTRACT FROM AUTHOR]

We prove that the Berezin transform is L p regular on a large class of domains in C n and not L 2 regular on the Hartogs triangle. [ABSTRACT FROM AUTHOR]

For the open unit disc D in the complex plane, it is well known that if φ ∈ C(D) then its Berezin transform ~ φ also belongs to C(D). We say that D is BC-regular. In this paper we study BC-regularity of some pseudoconvex domains in Cn and show that the boundary geometry plays an important role. We also establish a relationship between the essential norm of an operator in a natural Toeplitz subalgebra and its Berezin transform. [ABSTRACT FROM AUTHOR]

Let 1 ≤ q ≤ (n−1). We first show that a necessary condition for a Hankel operator on (0,q−1)-forms on a convex domain to be compact is that its symbol is holomorphic along q-dimensional analytic varieties in the boundary. Because maximal estimates (equivalently, a comparable eigenvalues condition on the Levi form of the boundary) turn out to be favorable for compactness of Hankel operators, this result then implies that on a convex domain, maximal estimates exclude analytic varieties from the boundary, except ones of top dimension (n−1) (and their subvarieties). Some of our techniques apply to general pseudoconvex domains to show that if the Levi form has comparable eigenvalues, or equivalently, if the domain admits maximal estimates, then compactness and subellipticity hold for forms at some level q if and only if they hold at all levels. [ABSTRACT FROM AUTHOR]

Let Ω be a C2-smooth bounded pseudoconvex domain in Cn for n≥2 and let φ be a holomorphic function on Ω that is C2-smooth on the closure of Ω. We prove that if Hφ¯ is in Schatten p-class for p≤2n then φ is a constant function. As a corollary, we show that the ∂¯-Neumann operator on Ω is not Hilbert-Schmidt. [ABSTRACT FROM AUTHOR]

Let Ω be a bounded convex Reinhardt domain in C2 and ϕ∈C(Ω¯). We show that the Hankel operator Hϕ is compact if and only if ϕ is holomorphic along every non-trivial analytic disc in the boundary of Ω. [ABSTRACT FROM AUTHOR]

Let Ω = Ω1 \ Ω2, where Ω1 and Ω2 are two smooth bounded pseudoconvex domains in Cn, n ≥ 3, such that Ω2 ⊂ Ω1. Assume that the ∂- Neumann operator of Ω1 is compact and the interior of the Levi-flat points in the boundary of Ω2 is not empty (in the relative topology). Then we show that the Hankel operator on Ω with symbol Φ,HΩ Φ , is compact for every Φ ∈ C(Ω) but the ∂-Neumann operator on Ω is not compact. [ABSTRACT FROM AUTHOR]