[Paper Review] A non-type (D) operator in c0

Orestes Bueno, B. F. Svaiter|arXiv (Cornell University)|Mar 11, 2011

Holomorphic and Operator Theory5 references5 citations

TL;DR

This paper constructs the first known example of a non-type (D) maximal monotone operator in the Banach space $c_0$, resolving a long-standing conjecture about the restriction of such operators to spaces containing isometric copies of $\ell^1$ or $L^1$. By leveraging Gossez's operator on $\ell^1$ and defining a dual operator on $c_0$, the authors prove that $c_0$ is not of type (D), and further show that any Banach space containing an isometric copy of $c_0$ also admits a non-type (D) operator.

ABSTRACT

Previous examples of non-type (D) maximal monotone operators were restricted to $\ell^1$, $L^1$, and Banach spaces containing isometriccopies 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 $c_0$.

Motivation & Objective

• To disprove the conjecture that non-reflexive Banach spaces of type (D) are restricted to those containing isometric copies of $\ell^1$ or $L^1$.
• To construct a concrete example of a maximal monotone operator on $c_0$ that is not of type (D).
• To establish that the property of admitting non-type (D) operators is preserved under isometric embedding into larger Banach spaces.

Proposed method

• Define a linear, continuous, anti-symmetric operator $G: \ell^1 \to \ell^\infty$ via $G(x)_n = \sum_{i=n+1}^\infty x_i - \sum_{i=1}^{n-1} x_i$, which is known to be maximal monotone and injective.
• Construct a dual operator $T: c_0 \rightrightarrows \ell^1$ by setting $T(x) = \{ y \in \ell^1 \mid -G(y) = x \}$, which is point-to-point and maximal monotone on $c_0$.
• Prove that $T$ is not of type (D) by showing that its Gossez monotone closure $\widetilde{T}$ contains a point $(x^{**}, x^*) \in X^{**} \times X^*$ that cannot be approximated by bounded nets in $\operatorname{Gr}(T)$ in the $\sigma(X^{**}, X^*) \times$ strong topology.
• Use the existence of a net $\{x^\tau\}$ in the bidual such that $(x^\tau, \tau \widetilde{y}) \in \widetilde{T}$ for $\widetilde{y} \in \ell^1$ with $\langle \widetilde{y}, e \rangle > 0$, and show that distinct extensions exist, violating uniqueness and implying non-type (D) status.
• Apply a general extension theorem to show that if $X$ admits a non-type (D) operator, then any Banach space $\Omega$ containing an isometric copy of $X$ also admits such an operator.
• Use the duality between $c_0$ and $\ell^1$ and the structure of the dual space to verify that the constructed operator $T$ satisfies the necessary inequalities for non-type (D) status.

Experimental results

Research questions

• RQ1 Does there exist a non-type (D) maximal monotone operator in $c_0$, a space not containing $\ell^1$ isometrically?
• RQ2 Can the conjecture that all non-reflexive spaces without $\ell^1$-copies are of type (D) be true, particularly for $c_0$?
• RQ3 Is the property of admitting a non-type (D) operator preserved under isometric embedding into larger Banach spaces?
• RQ4 Can Gossez's monotone closure fail to be the unique maximal monotone extension in non-reflexive spaces?
• RQ5 What is the role of the dual space structure in determining whether a maximal monotone operator is of type (D)?
• RQ6 Is there a systematic way to construct non-type (D) operators in non-reflexive Banach spaces?

Key findings

• The paper constructs a linear, maximal monotone operator $T: c_0 \rightrightarrows \ell^1$ that is not of type (D), providing the first such example in $c_0$.
• The operator $T$ is defined as the preimage under $-G$ of elements in $c_0$, where $G$ is Gossez's known non-type (D) operator on $\ell^1$.
• It is shown that $T$ is not of type (D) because its Gossez monotone closure $\widetilde{T}$ contains a point $(x^{**}, x^*)$ that cannot be approximated by bounded nets in $\operatorname{Gr}(T)$ in the $\sigma(X^{**}, X^*) \times$ strong topology.
• The existence of distinct maximal monotone extensions of $T$ to $\ell^\infty$ is demonstrated via a net $\{x^\tau\}$, which implies non-uniqueness of extension and thus non-type (D) status.
• The paper proves that any Banach space $\Omega$ containing an isometric copy of $c_0$ admits a non-type (D) maximal monotone operator, extending the result beyond $c_0$ itself.
• The result refutes the conjecture that non-reflexive spaces without $\ell^1$-copies (such as $c_0$) are necessarily of type (D), showing that such spaces can still host non-type (D) operators.