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On maximally accretive operators in the plane. (English) Zbl 0594.47045
Approximation theory and functional analysis, Anniv. Vol., Proc. Conf., Oberwolfach 1983, ISNM 65, 109-116 (1984).
[For the entire collection see Zbl 0537.00009.]
M. G. Crandall and T. M. Ligget [Trans. Am. Math. Soc. 160, 263-278 (1971; Zbl 0226.47037)] showed that for the plane endowed with the \(\ell_ p\) norm, \(1\leq p\leq \infty\), the class of m-accretive operators coincides with the class of maximally accretive operators exactly when \(p=1,2\), or \(\infty\). In this paper, the authors prove the following theorem:
Let X be a real, 2-dimensional, normed vector space with a strictly convex and smooth norm. If every accretive operator in \(X\times X\) has an m-accretive extension then the norm generates an inner product.
The proof is based on constructing certain accretive operators associated to two pairs of conjugate subspaces. The convexity of the closure of the domain of an m-accretive operator plays an important role.
Reviewer: F.Bernis

47H06 Nonlinear accretive operators, dissipative operators, etc.