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The goal of this paper is to provide complete analogues of our previous results from [17,27] in the unit disk and unit ball in context of more complicated tubular domains over symmetric cones. This paper consists of two separate parts. The first part is devoted to new assertions on integral Bergman-type operators acting on some new Bergman type spaces in products (or semi-products) of tubular domains over symmetric cones and the second one with related new analytic multifunctional spaces in tubular domains over symmetric cones. Previously it was noted these two topics are related in simpler domains in the unit disk and in the unit ball (see [17, 26, 27]). Parallel assertions are also valid in context of general bounded strictly pseudoconvex domains with smooth boundary (see [29, 32]). To summarize this paper and [29, 32] show in most typical Siegel domains such as bounded strongly pseudonconvex domains with smooth boundary and unbounded tubular domains over symmetric cones main assertions from [17,27] are also valid. It is a fundamental fact of the theory of analytic Bergman function

Bergman-type integral operators on semiproducts of the tube domains over symmetric cones and multifunctional analytic spaces