Operator Ranges in Banach Spaces, I

Abstract

AbstractLet 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, lp (1 < p < ∞), Lq (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 L1, 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.