Conditions ensuring T ⎺¹ (Y ) ⊂ Y

Abstract

The following theorem is the main result of the paper: Let X be a complex Banach space and 𝑇 ∈ L(X). Suppose that 0 lies at the unbounded component of the set of those λ such that λI − 𝑇 is a Fredholm operator. Let Y be a dense subspace of the dual space X′ and S be a closed operator from Y to X such that 𝑇 ′( Y ) ⊂ Y and 𝑇 S𝓎 = ST ′𝓎 for each 𝓎 ∈ Y . Then for each vector 𝓍 ∈ X′, 𝑇 ′𝓍 ∈ Y𝓍 if and only if 𝓍 ∈ Y .