Plichko, Anatolij M.; Popov, Mikhail M. Symmetric function spaces on atomless probability spaces. (English) Zbl 0715.46011 Diss. Math. 306, 85 p. (1990). The work under review is concerned with various more or less separate topics on symmetric function spaces. Here is a selection of what the authors have proved. Within preliminary sections (Sections 1-3) they provide a new proof of the Maharam’s theorem. In what follows, E stands for a symmetric function space over an atomless probability space. Section 4: Bases in nonseparable symmetric spaces. The authors prove that the Walsh system in E with absolutely continuous norm forms an Enflo- Rosenthal-Cesàro basis (Theorem 1). Further properties of the Walsh system in E are provided in Theorem 2; they depend upon the Boyd indexes. Section 5: Uncountable unconditional basis sequences and unconditional decompositions. Main result: For E with absolutely continuous norm, every subspace of E has a monotone unconditional projection decomposition provided that the upper Boyd index is finite and the lower one is greater than 1. This generalizes a result of W. B. Johnson and E. Odell [Compositio Math. 28, 37-49 (1974; Zbl 0282.46020)], where \(E=L_ p(\mu)\) with \(1<p<\infty.\) Section 6: Characters of compact Abelian groups. Theorem 3: The character group \(G'\) of an arbitrary compact Abelian group G can be decomposed into countably many subsets each of which is equivalent in every \(L_ p(G)\) \((1\leq p<\infty)\) to an orthonormal basis of a Hilbert space. Section 7: Almost periodic functions. The notion of a symmetric space of almost priodic functions is introduced by extending Basicovitch spaces and some of their generalizations, and the previous results are applied to these spaces. Section 8: Narrow operators. Narrow operator are characterized (another terminology: norm-sign-preserving operators) and a class of symmetric spaces on which there is no non-zero narrow operator is exhibited. Section 9: Narrow operators in \(L_ p(\mu)\), \(1\leq p<\infty\). Theorem 7: Let T be a narrow operator from \(L_ p(\mu)\) into itself \((1<p<\infty\), \(p\neq 2)\). Then for each \(\epsilon >0\) there is \(\delta_ p(\epsilon)>0\) with \(\| I+T\| \geq 1+\delta_ p(\| T\|)\) where I is the identity operator of \(L_ p(\mu)\). This was known previously for T a compact self-operator of \(L_ p(0,1).\) Section 10: Rich subspaces. Section 11: Isomorphic classification of \(L_ p(\mu)\)-spaces. Theorem 1: With \(0<p<1\), \(L_ p(\mu)\) and \(L_ p(\nu)\) are isomorphic iff the Maharam set of measures \(\mu\) and \(\nu\) coincide. Theorem 2: If \(1\leq p<\infty\), \(p\neq 2\), and the Maharam sets of \(\mu\) and \(\nu\) are distinct, then the Banach-Mazur distance between \(L_ p(\mu)\) and \(L_ p(\nu)\) is \(\geq 2\) if \(p=1\), and is \(\geq 1+\delta_ p(1)\) otherwise. Reviewer: T.Kubiak Cited in 16 ReviewsCited in 42 Documents MSC: 46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces Keywords:Maharam’s theorem; symmetric function space over an atomless probability space; Bases in nonseparable symmetric spaces; Walsh system; Enflo- Rosenthal-Cesàro basis; Boyd indexes; Uncountable unconditional basis sequences; unconditional decompositions; absolutely continuous norm; unconditional projection decomposition; Characters of compact Abelian groups; Almost periodic functions; Basicovitch spaces; Narrow operators; norm-sign-preserving operators; Rich subspaces; Isomorphic classification of \(L_ p(\mu )\)-spaces; Maharam sets; Banach-Mazur distance Citations:Zbl 0282.46020 PDFBibTeX XML