Geometry of Müntz spaces and related questions
For a given Banach space \(E\) of functions on \([0,1]\), a sequence of the form \(\{t^{\lambda_n}\} \subset E\), \(0=\lambda_0<\lambda_1< \ldots\), is called a \textit{Müntz sequence}, and the subspace spanned in \(E\) by such a sequence is called a \textit{Müntz space}. According to the famous Müntz theorem, if \(E=C[0,1]\) or \(E=L_p[0,1]\), \(1 \leq p < \infty\), and \(\sum \lambda_n^{-1} < \infty\), the corresponding Müntz space does not coincide with \(E\). In this case, the study of the properties of this Müntz space (which is the main objective of the book under review) appears to be an interesting and highly non-trivial task. The Müntz sequence turns out to be minimal, but not necessarily uniformly minimal or a basis. For a Müntz sequence in \(E=C[0,1]\) or \(E=L_p[0,1]\), \(1 \leq p < \infty\), to be uniformly minimal is equivalent to be a basic sequence and is, in turn, equivalent to lacunarity in the following sense: \(\inf_k \lambda_{k+1}/\lambda_k > 1\). In this case (and, more generally, for quasi-lacunary sequences of \(\lambda_k\)), the Müntz space is isomorphic to \(c_0\) if \(E=C[0,1]\) or to \(\ell_p\) if \(E=L_p[0,1]\). There are at least two isomorphism classes of Müntz spaces in \(E=C[0,1]\) and infinitely many isometry classes. The book consists of two parts. The first one, ``Subspaces and sequences in Banach spaces'', has five chapters: Disposition of subspaces; Sequences in normed spaces; Isomorphisms, isometries and embedddings; Spaces of universal disposition; Approximation properties. Although this part gives some necessary background to the study of Müntz spaces, the main goal is to give an introduction to the disposition of subspaces language and technique in the framework of Banach space theory and to present, among other things, some of his results obtained in the sixties and seventies and published in Russian. The second part, ``On the geometry of Müntz sequences'', consists of seven chapters: Coefficient estimates and the Müntz theorem; Classification and elementary properties of Müntz sequences; More on the geometry of Müntz sequences and Müntz polynomials; Operators of finite rank and bases in Müntz spaces; Projection types and the isomorphism problem for Müntz spaces; The classes \([\mathcal M], \mathcal A, \mathcal P\) and \(\mathcal P_\epsilon\); Finite dimensional Müntz limiting spaces in \(C\). There are a lot of open problems in the text. For example, it is unknown if a Müntz space in \(C[0,1]\) always has a basis, and even for the span of \(\{t^{n^2}\}_{n\in \mathbb N}\) in \(C[0,1]\) the existence of a basis is an open problem.
scientific article; zbMATH DE number 5012644