A non-type (D) operator in $$c_0$$
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A non-type (D) operator in c0 Orestes Bueno · B. F. Svaiter
Received: 11 March 2011 / Accepted: 3 September 2011 / Published online: 20 March 2013 © Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013
Abstract Previous examples of non-type (D) maximal monotone operators were restricted to 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 c0 . Keywords
Maximal monotone · Type (D) · Banach space · Extension · Bidual
Mathematics Subject Classification (2000)
47H05 · 49J52
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)},
Dedicated to Professor J. M. Borwein on the occasion of his 60th birthday. O. Bueno · B. F. Svaiter (B) Instituto de Matématica Pura e Aplicada (IMPA), Estrada Dona Castorina 110, Rio de Janeiro, RJ, CEP 22460-320, Brazil e-mail: [email protected] O. Bueno e-mail: [email protected]
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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 , T −1 (v) = {u ∈ U | v ∈ T (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. Let X be a real Banach space. We use the notation X ∗ for the topological dual of X . From now on X is identified with its canonical injection into X ∗∗ = (X ∗ )∗ and the duality product in X × X ∗ will be denoted by ·, · , x, x ∗ = x ∗ , x = x ∗ (x), x ∈ X, x ∗ ∈ X ∗ . A point-to-set operator T : X ⇒ X ∗ (respectively T : X ∗∗ ⇒ X ∗ ) is monotone, if x − y, x ∗ − y ∗ ≥ 0, ∀(x, x ∗ ), (y, y ∗ ) ∈ Gr(T ), (resp. x ∗ − y ∗ , x ∗∗ − y ∗∗ ≥ 0, ∀(x ∗∗ , x ∗ ), (y ∗∗ , y ∗ ) ∈ Gr(T )), and it is maximal monotone if it is monotone and maximal in the family of monotone operators in X × X ∗ (resp. X ∗∗ × X ∗ ) with respect to the order of inclusion of the graphs. We denote c0 as the space of real sequences converging to 0 and ∞ as the space of real bounded sequences, both endowed with the sup-norm
(xk )k ∞ = sup |xk |, k∈N
and 1 as the space of absolutely summable real sequences, endowed with the 1-norm,
(xk )k 1 =
∞
|xk |.
k=1
The dual of c0 is identified with 1 in the following sense: for y ∈ 1 y(x) = x, y =
∞
xi yi , ∀x ∈ c0 .
i=1
Likewise, the dual of 1 is identified with ∞ . It is well known that c0 (as well as 1 , ∞ , etc.) is a non-reflexive Banach space. Let X be a non-reflexive real Banach space and T : X ⇒ X ∗ be maximal monotone. Since X ⊂ X ∗∗ , the point-to-set operator T can also be regarded as an operator from
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A non-type (D) operator in c0
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