  In this note we describe conditions under which, in idempotent functional
analysis, 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. Our results provide a general description of a class of
subsemimodules of the semimodule of all bounded functions with values in the
Max-Plus algebra where some kind of kernel theorem holds, thus addressing an
open problem posed by J. Gunawardena. 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. In this work, a rather general case of semimodules over boundedly
complete idempotent semirings is considered.
