A non-type (D) operator in $$c_0$$

Abstract

Previous examples of non-type (D) maximal monotone operators were restricted to ℓ 1 , L 1 , and Banach spaces containing isometric copies of these spaces. This fact led to the conjecture that non-type (D) operators were restricted to this class of Banach spaces. We present a linear non-type (D) operator in c 0 . keywords: maximal monotone, type (D), Banach space, extension, bidual. 1 Introduction Let U, V arbitrary sets. A point-to-set (or multivalued) operator T : U ⇒ V is a map T : U → P(V ), where P(V ) is the power set of V . Given T : U ⇒ V , the graph of T is the set Gr(T ) := {(u, v) ∈ U × V | v ∈ T (u)}, the domain and the range of T are, respectively, dom(T ) := {u ∈ U | T (u) = ∅}, R(T ) := {v ∈ V | ∃u ∈ U, v ∈ T (u)} and the inverse of T is the point-to-set operator T −1 : V ⇒ U,A point-to-set operator T : U ⇒ V is called point-to-point if for every u ∈ dom(T ), T (u) has only one element. Trivially, a point-to-point operator is injective if, and only if, its inverse is also point-to-point.

Recent Progress on Monotone Operator Theory

“…They also showed that each Banach space that contains an isometric (isomorphic) copy of c 0 is not of type (D) in [27]. Example 2.12(xi) recaptures their result, while Example 2.12(vi)&(viii) provide a negative answer to Simons' [61, Problem 22.12].…”

“…(xiii) Moreover, G is a unique maximally monotone operator that is not of type (D), but G is isomorphically of type (BR Bueno and Svaiter had already showed that T e is not of type (D) in [27]. They also showed that each Banach space that contains an isometric (isomorphic) copy of c 0 is not of type (D) in [27].…”

Construction of Pathological Maximally Monotone Operators on Non-reflexive Banach Spaces

“…Bueno and Svaiter also showed that T e is not of type (D) in [12]. They also showed that each Banach space that contains an isometric (isomorphic) copy of c 0 is not of type (D) in [12] Thus, in lattices such as c 0 , c and C[0, 1] only discontinuous linear monotone operators can fail to be of type (D). This subtlety escaped the current authors for fifteen years.…”

Recent progress on Monotone Operator Theory

“…Theorem 2.10 below allows us to construct various maximally monotone operators -both linear and nonlinear -that are not of type (D). The idea of constructing the operators in the following fashion is based upon [2, Theorem 5.1] and was stimulated by [29].…”

“…(xiii) Moreover, G is a unique maximally monotone operator that is not of type (D), but G is isomorphically of type (BR Bueno and Svaiter had already showed that T e is not of type (D) in [29]. They also showed that each Banach space that contains an isometric (isomorphic) copy of c 0 is not of type (D) in [29]. Example 2.12(xi) recaptures their result, while Example 2.12(vi)&(viii) provide a negative answer to Simons' [65,Problem 22.12].…”