Analysis of Operators on Idempotent Semimodules

Abstract

In this chapter we study endomorphisms, or linear operators, on semimodules of functions that range in idempotent semirings. Here the specific nature of idempotent analysis exhibits itself in the fact that each linear operator on such a semimodule is an integral operator, that is, has the form $$ \left( {Bh} \right)\left( x \right) = \int {^ \oplus b\left( {x,y} \right) \odot h\left( y \right)d\mu \left( y \right) = \mathop {\inf }\limits_y \left( {b\left( {x,y} \right) \odot h\left( y \right)} \right)} $$ for some idempotent integral kernel b(x,y). In §2.1 we give necessary and sufficient conditions for this function to specify a continuous operator. We give a characterization of weak and strong convergence of operator families in terms of kernels and then, in §2.2, we describe two important operator classes— invertible and compact operators—in the same terms. Here another specific feature of the semialgebra of idempotent linear operators is important—the supply of invertible operators is very small; namely, the group of invertible operators is generated by the diagonal operators and by the homomorphisms of the base. Hence, this group consists of idempotent analogs of weighted translation operators. It follows that all automorphisms of the operator semialgebra are inner automorphisms.