Invariant sublattices for positive operators (Q2381834)

The main part of the paper is a detailed discussion of some examples of positive operators, including lattice homomorphisms, on classical Banach lattices, which do not have nontrivial closed invariant (vector) sublattices. This part is an extended version of an earlier paper by the same authors [in:\ Proceedings of the conference ``Positivity IV -- Theory and Applications'', Dresden (2005), 73--77 (2006; Zbl 1115.47035)]. The examples are interesting, but the discussion is, alas, rather careless [see, e.g., the formulations of Lemmas 3.2 and 3.3, and a lapse concerning the Liouville numbers on p.~59]. The paper also contains some results on general Banach lattices and positive operators on them concerned with arbitrary (not necessarily closed) sublattices and their invariance (Section~2) as well as two open questions (Section ~5). Reviewer's remark: Proposition 2.1 is a special case of each of the following two results: Exercise II.5(e) in [\textit{H. H. Schaefer}, ``Banach lattices and positive operators '' (1974; Zbl 0296.47023)] and Lemma~7 in [\textit{Y. A. Abramovich} and \textit{Z. Lipecki}, Math. Proc. Camb. Philos. Soc. 108, No. 1, 79--87 (1990; Zbl 0751.46009)]. From the latter result, the Baire category theorem and a well-known characterization of finite-dimensional Banach spaces as locally compact ones, Proposition~2.4 and Corollary~2.5 also follow. In the same way, one can answer Question~5.1 in the affirmative and generalize a part of Theorem~2.6 as follows: Given a topological vector lattice \(X\), a \(K_\sigma\)-subset \(K\) of \(X\) and continuous maps \(f_1,f_2,\dots\) of \(X\) into itself, there exists a sublattice \(Y\) of \(X\) such that: (1) \(Y\) is a \(K_\sigma\)-subset of \(X\), (2) \(K\subset Y\), (3) \(f_i(Y)\subset Y\) for each \(i\).