On a convexity property of the range of a maximal monotone operator

An example is given which shows that the closure of the range of a maximal monotone operator from a (nonreflexive) Banach space into its dual is not necessarily convex. Introduction. Let A be a real Banach space with dual X* and let T: X -* 2X be a maximal monotone operator with domain D{T) and range R{T). In general R{T) is not a convex set (cf. [4]) but it is known that when X is reflexive, the (norm) closure of R{T) is convex (cf. [5]). Without reflexivity, the convexity of cl R{T) is still true when T is the subdifferential of a lower semicontinuous proper convex function (cf. [1]), or more generally, when the associated monotone operator Tx: X" —> 2X is maximal (cf. [2] where the proof is given under a slightly stronger assumption). Here Tx denotes the operator whose graph is defined by gr7f = {(x",x*) E X"xX*;3 a net {Xj,x*) E grFwith Xj bounded,xt -* x" weak" and x-> x* in norm}. {X is identified as usual to a subspace of its bidual X**.) The question was raised some years ago as to whether or not the convexity of cl R{T) holds in general. In this note we answer this question negatively. We exhibit a (everywhere defined and coercive) maximal monotone operator from /' to 2/0° whose range has not a convex closure. Our construction is based on a result of [3]. Example. Let A: lx -> /°° be the bounded linear operator defined by 00 {Ax)n = 2l xmamn m=\ for x = {xx,x2,...) E /', where a■ = 0 if m = n, amn = -1 if n > m and am„ = +1 if n 2R is given by s{t) = -I if t 0. For A > 0, the mapping XJ + A is clearly maxReceived by the editors March 31, 1975. A MS (MOS) subject classifications (1970). Primary 47H05; Secondary 46B10, 35J60.