Magnifying elements in semigroups of linear transformations with invariant subspaces

Abstract

Let L(V) be the linear transformation semigroup on a vector space V. It is well-known that L(V) contains left [right] magnifying elements if and only if the dimension of V is infinite. In case its dimension is infinite, α ϵ L(V) is left magnifying if and only if it is surjective and not injective and it is right magnifying if and only if it is injective and not surjective. To generalize this result, let W be a subspace of V and S(V, W) = {f ∈ L(V) | (W) f ⊆ W}. Then S(V, W) is a subsemigroup of L(V) and if W = V, then S(V, W) = S(V, V) = L(V). Our purpose in this paper is to give necessary and sufficient conditions for elements in S(V, W) to be left [right] magnifying.