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In this paper we describe, via the Laplace transformation of analytic functionals, a pre-dual to the function algebra A −∞ (D) (D being either a bounded C 2 -smooth convex domain in C N (N > 1), or a bounded convex domain in C) as a space of entire functions with certain growth. A possibility of representation of functions from the pre-dual space in a form of Dirichlet series with frequencies from D, is also studied.

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We study weighted composition operators acting between Fock spaces. The following results are obtained:(i) Criteria for the boundedness and compactness.(ii) Characterizations of compact differences and essential norm.(iii) Complete descriptions of path connected components and isolated points of the space of composition operators and the space of nonzero weighted composition operators.

Abstract. In this paper we present the following results: a description, via the Laplace transformation of analytic functionals, of the dual to the (DFS)-

In the space A −∞ (D) of functions of polynomial growth, weakly sufficient sets are those such that the topology induced by restriction to the set coincides with the topology of the original space. Horowitz, Korenblum and Pinchuk defined sampling sets for A −∞ (D) as those such that the restriction of a function to the set determines the type of growth of the function. We show that sampling sets are always weakly sufficient, that weakly sufficient sets are always of uniqueness, and provide examples of discrete sets that show that the converse implications do not hold.

We characterize all selfadjoint as well as all unitary anti-linear weighted composition operators acting on the Fock space F 2 (C n ). Then we obtain a complete description of anti-linear weighted composition operators that are conjugations. These results allow us to determine which bounded linear weighted composition operators can be complex symmetric on F 2 (C n ).

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