The problem of central orthomorphisms in a class of \(F\)-lattices
This paper is concerned with the following problem: \(( P)\) Characterize those Archimedean vector lattices \(E\) for which every order bounded orthomorphism on \(E\) is a central operator, that is, an operator satisfying \(|T | \leq \lambda I_E\) for some positive real number \(\lambda\), where \(I_E\) is the identity on \(E\). Let \(Z(E)\) denote the central operators on \(E\) and let \(\mathrm{Orth}(E)\) denote the orthomorphisms on \(E\). In general, we have \(Z(E) \subset \mathrm{Orth}(E)\). In [Compos. Math. 35, 225--238 (1977; Zbl 0381.47021)], \textit{A. W. Wickstead} showed that, if \(E\) is a Banach lattice, then \(Z(E) = \mathrm{Orth}(E)\). The introduction of the paper contains a brief history of the problem \((P)\). Recall that a subadditive function \(x \mapsto ||x||\) defined on a linear space \(X\) such that \(||x|| = 0\) if and only if \(x = 0\) and, for every \(x \in X\), the function \(t \mapsto ||tx||\) is continuous at zero, is called an \(F\)-norm on \(X\). If \(X\) is complete with respect to \(||\cdot||\), then \(X\) is called an \(F\)-space. An \(F\)-space \(X\) is called locally bounded at \(0\) if there exists an open ball in \(X\) centered at \(0\) which is topologically bounded. If the vector lattice \(E\) is endowed with a monotone \(F\)-norm \(||\cdot||\) under which \(E\) is complete, then \((E, ||\cdot||)\) is said to be an \(F\)-lattice. Every Banach lattice is an \(F\)-lattice. The main result of the paper is the following. Theorem. If \((E, ||\cdot||)\) is a locally bounded \(F\)-lattice, then \(\mathrm{Orth}(E) = Z(E)\). The preceding result is then applied to Nakano spaces. Let \((p_n )\) be a fixed sequence of numbers in the interval \((0, 1)\). Consider the space of sequences \(l^{(p_n )}\), where \[ l^{(p_n )}= \left\{(x_n ) :\sum^{\infty}_{n=1}|x_n |{}^{p_n} < \infty\right\}, \] endowed with the \(F\)-norm \(\displaystyle ||(x_n )|| =\sum^{\infty}_{ n=1} |x_n |{}^{p_n}\). The next result answers problem \((P)\) for the class of Nakano spaces. Theorem. Let \((p_n )\) be a fixed sequence in \((0, 1)\). For the Nakano \(F\)-lattice \(E = l^{(p_n )}\), the following are equivalent: {\parindent=8mm \begin{itemize} \item[(i)] \(\mathrm{Orth}(E) = Z(E)\). \item [(ii)] \(\inf p_n > 0\). \item [(iii)] \(E\) is locally bounded. \end{itemize}} In the final part of the paper, the results are generalized to Musielak-Orlicz sequence spaces. In particular, if \(\Phi = (\phi_n )\) denotes a sequence of increasing, continuous Orlicz functions which contains a subsequence with a certain condition \((\Delta_n\), in the paper), then \(\mathrm{Orth}(l^\Phi) = Z(l^\Phi)\).
scientific article; zbMATH DE number 6418462
scientific article