Operator Ranges in Banach Spaces,I

Let X be a Banach space. A subspace L of X is called an operator range if there exists a continuous linear operator T defined on some Banach space Y and such that TY = L. If Y = X then L is called an endomorphism range. The paper investigates operator ranges under the following perspectives: (1) Existence (Section 3), (2) Inclusion (Section 4), and (3) Decomposition (Section 5). Section 3 considers the existence in X of operator ranges satisfying certain conditions. The main result is the following: if X and Fare separable Banach spaces and T : Y → X is a continuous operator with nonclosed range, then there exists a nuclear operator R:Y→X such that T + R is injective and has nonclosed dense range (Theorem 3.2). Section 4 seeks to determine conditions under which every nonclosed operator range in a Banach space is contained in the range of some injective endomorphism with nonclosed dense range. Theorem 4.3 contains a sufficient condition for this. Examples of spaces satisfying this condition are c₀, l_p (1 < p < ∞), L_q (1 < q < 2) and their quotients. In particular, this answers a question posed by W. E. Longstaff and P. Rosenthal (Integral Equations and Operator Theory 9 , (1986), 820-830. Section 5 discusses the possibility of representing a given dense nonclosed operator range as the sum of a pair L_1, L₂ of operator ranges with zero intersection in the cases where (a) L₁ and L₂ are dense, (b) L₁ and L₂ are closed. The results generalize corresponding results, for endomorphisms in Hilbert space, of J. Dixmier (Bull. Soc. Math. France 77 (1949), 11-101 and P. A. Fillmore and J. P. Williams (Adv. Math. 7 (1971), 254-281. A final section is devoted to open problems.