Nuclear Semimodules and Kernel Theorems in Idempotent Analysis. an Algebraic Approach

In this note we describe conditions under which, in idempotent functional analysis (see [1–3]), linear operators have integral representations in terms of idempotent integral of V. P. Maslov. We define the notion of nuclear idempotent semimodule and describe idempotent analogs of the classical kernel theorems of L. Schwartz and A. Grothendieck (see, e.g., [4–6]). In [11], for the idempotent semimodule of all bounded functions with values in the Max-Plus algebra, there was posed a problem of describing the class of subsemimodules where some kind of kernel theorem holds. Some of the results obtained here can be regarded as possible versions of an answer to this question. Previously, some theorems on integral representations were obtained for a number of specific semimodules consisting of continuous or bounded functions taking values mostly in the Max-Plus algebra (see, e.g., [7–10, 1, 3]). In this work (and further publications), a very general case of semimodules over boundedly complete idempotent semirings is considered. This note continues the series of publications [1–3]; we use the notation and terminology defined in those articles. 1. Functional semimodules. Idempotent analysis is based on replacing the number fields by idempotent semifields and semirings. In other words, a new set of basic associative operations (a new addition ⊕ and new multiplication ⊙) replaces the traditional arithmetic operations. These new operations must satisfy all axioms of semifield or semiring; also, the new addition must be idempotent, i.e., satisfy x ⊕ x = x for all x belonging to the semifield or semiring (see, e.g., [7–11]). A typical example is the semifield R max = R {−∞}, which is known as the Max-Plus algebra. This semifield consists of all real numbers and an additional element −∞, denoted by 0, that is the zero element of R max ; the operations are defined by x ⊕ y = max{x, y} and x ⊙ y = x + y. The unit element 1 of R max coincides with the usual zero. One can find a lot of nontrivial examples of idempotent semirings and semifields, e.g., in [3, 7–11]. An idempotent semimodule over an idempotent semiring K is an additive commutative idempotent semigroup, with the addition operation denoted by ⊕ and the zero element denoted by 0, such that a product k ⊙ x is defined, for all k in K and x in the semimodule, in such a way that the usual rules are satisfied. For …