In this paper we completely describe the automorphism group of the Hartogs type domains over bounded classical symmetric domains by using the ball characterization theorem about noncompact automorphism groups. As an application, we give plenty of examples of non-homogeneous domains in ℂn, for which the real dimension of the automorphism group is less than n2 - 1.

Let f and g be functions, not identically zero, in the Fock space F 2 α of C n . We show that the product T f T g of Toeplitz operators on F 2 α is bounded if and only if f (z) = e q(z) and g(z) = ce −q(z) , where c is a nonzero constant and q is a linear polynomial.

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Abstract. We compute the explicit formula of the Bergman kernel for a nonhomogeneous domain {(z 1 , z 2 ) ∈ C 2 : |z 1 | 4/q 1 + |z 2 | 4/q 2 < 1} for any positive integers q 1 and q 2 . We also prove that among the domains