The Factor Decomposition Theorem of Bounded Generalized Inverse Modules and Their Topological Continuity

In this paper we obtain a Douglas type factor decomposition theorem about certain important bounded module maps. Thus, we come to the discussion of the topological continuity of bounded generalized inverse module maps. Let X be a topological space, x → T _x : X → L(E) be a continuous map, and each R(T _x ) be a closed submodule in E, for every fixed x ∈ X. Then the map is continuous if and only if is locally bounded, where is the bounded generalized inverse module map of T _x . Furthermore, this is equivalent to the following statement: For each x ₀ in X, there exists a neighborhood U ₀ at x ₀ and a positive number λ such that where σ(T) denotes the spectrum of operator T. Project supported by NSF of China “Maximal regularity for vector-valued boundary problems”(10571099) and NSF of China(10571003)