On non-enlargeable positive sets
(\({\mathcal B}, \lfloor.,.\rfloor\)) is called a \textit{symmetrically self-dual space} (SSD space) if \(\mathcal B\) is a non-zero real vector space and \(\lfloor.,.\rfloor: {\mathcal B} \times {\mathcal B} \rightarrow \mathbb{R}\) is a symmetric bilinear form. The bilinear form \(\lfloor.,.\rfloor\) induces the quadratic form defined by \(q(b):=\frac{1}{2}\lfloor b,b \rfloor\), \(b \in {\mathcal B}\). A nonempty subset \(A\) of an SSD space \({\mathcal B}\) is said to be \textit{\(q\)-positive} if one has \(q(a-c) \geq 0\) for all \(a,c \in A\). For a \(q\)-positive subset \(A\) of an SSD space \(\mathcal B\) and for \(\epsilon \geq 0\), the \(\epsilon\)-enlargement of \(A\) is defined by \(A^{\epsilon}:=\{x \in {\mathcal B}: q(x-a) \geq -\epsilon\) for each \(a \in A\}\). We say that \(A\) is \textit{non-enlargeable} if \(A=A^{\epsilon}\) for every \(\epsilon \geq 0\). The authors present characterizations of non-enlargeable positive sets in SSD spaces. An SNL-map (SNL, acronym for ``symmetric nonnexpansive linear'') on \(\mathcal B\) is a linear map \(L:{\mathcal B} \rightarrow {\mathcal B}^*\) such that \(\| L \| \leq 1\) and \(\langle b,Lc\rangle = \langle c,Lb\rangle\) for all \(b,c \in {\mathcal B}\). Here, \({\mathcal B}^*\) is the dual space of \(\mathcal B\) and \(\langle . , . \rangle\) denotes the dual pairing. A Banach SNL space is a nonzero real Banach space \(\mathcal B\) together with an SNL map \(L\). Let \(\mathcal B\) be a Banach SNL space. Define \(q_L\) on \(\mathcal B\) by \(q_L(b):=\frac{1}{2}\langle b,Lb\rangle\), \(b \in {\mathcal B}\). Positive sets and non-enlargeable sets are now defined in terms of the map \(q_L\). Characterizations of non-enlargeable positive sets in Banach SNL spaces are also presented.
scientific article; zbMATH DE number 6657274
scientific article