Differential operators of infinite order appear naturally in various fields of mathematics and physics. They play crucial roles in a variety of researches in those fields. The spaces of entire functions with several complex variables of prescribed orders endowed with natural topologies and continuous homomorphisms between two of these spaces are our main interests. We prove that such homomorphisms are represented by infinite-order (or finite-order) linear differential operators with entire coefficients satisfying some growth conditions. Conversely, such operators induce continuous homomorphisms. Thus one can manipulate the continuous homomorphisms in the spaces of entire functions by using linear differential operators of infinite order with the aid of their symbol calculus.