Local monotonicity structure of Calderón-Lozanovskiĭ spaces.
Let \((E, \|\cdot \|_E\)) be a Köthe space of real measurable functions of a complete and \(\sigma\)-finite measure space. In this paper, the authors study the existence and properties of the so-called points of upper-lower monotonicity (UM-point, LM-point), i.e., \(x \in E^+\) is UM if for all \(y \in E^+ \backslash \{0\}\), \(\| x \|_E < \| x+y\|_E\) and \(x \in E^-\) is LM if for all \(y \in E^+\), \(y \leq x\), \(y \not=x\) we have \(\| y \|_E < \| x \|_E\). If every positive function of the unit sphere is UM (or LM), then \(E\) is called strictly monodrone. A similar concept, that of upper (lower) local uniform monotonicity (ULUM), (LLUM) is presented. A number of results are given in which the classical Orlicz spaces play an important role. We quote the following: If \(\phi\) has \(\Delta_2^E\), \(0< \phi < \infty\), then \(x \in E^+_\phi\) is ULUM whenever \(\phi \circ x\) is a ULUM point of \(E\).
scientific article; zbMATH DE number 2124641
scientific article