Proceedings of the Royal Society of Edinburgh, 146A, 325–336, 2016 DOI:10.1017/S0308210515000451 Domains of accretive operators in Banach spaces Jes´ us Garcia-Falset and Omar Mu˜ niz-P´ erez Departament d’An` alisi Matem`atica, Universitat de Val` encia, Dr Moliner 50, 46100 Burjassot, Val` encia, Spain (garciaf@uv.es; omar.muniz@uv.es) Simeon Reich Department of Mathematics, Technion – Israel Institute of Technology, 32000 Haifa, Israel (sreich@tx.technion.ac.il) (MS received 28 April 2014; accepted 9 January 2015) Let D(A) be the domain of an m-accretive operator A on a Banach space E. We provide sufficient conditions for the closure of D(A) to be convex and for D(A) to coincide with E itself. Several related results and pertinent examples are also included. Keywords: accretive operator; Banach space; duality mapping; fixed point; Kadec–Klee property; resolvent 2010 Mathematics subject classification: Primary 46T99; 47H04; 47H06; 47H10 1. Introduction Let (E, ‖·‖) be a real Banach space and let E ∗ denote its dual. We call a mapping A : E → 2 E an operator on E. The domain of A, that is, the set {x ∈ E : Ax = ∅}, is denoted by D(A), and its range by R(A). An operator A on E is said to be accretive if the inequality ‖x − y + λ(u − v)‖  ‖x − y‖ holds for all λ> 0, x, y ∈ D(A), u ∈ Ax and v ∈ Ay. If, in addition, R(I + λA) (that is, the range of the operator I + λA) is precisely E for one (hence for all) λ> 0, then A is called m-accretive. If D(A) ⊆  λ>0 R(I + λA), then A is said to satisfy the range condition. Accretive operators were independently introduced more than forty-five years ago by Browder [4] and Kato [12]. Those accretive operators that either are m-accretive or satisfy the range condition play an important role in the study of differential equations in Banach spaces. On the one hand, it is well known that if E is a reflexive Banach space and A : E → 2 E is an m-accretive operator, then the initial-value problem u ′ (t)+ Au(t) ∋ 0, u(0) = x 0 ,  (1.1) has a unique strong solution whenever x 0 ∈ D(A) (see, for example, [1, theo- rem 4.5]). Therefore, it is of interest to know the conditions under which the domain of an m-accretive operator coincides with the whole space. 325 c  2016 The Royal Society of Edinburgh