Linear and complex analysis. Problem book. 199 research problems
type: Item
instance of: scholarly article
title: Linear and complex analysis. Problem book. 199 research problems
zbMATH Open document ID: 0545.30038
published in: Lecture Notes in Mathematics
publication date: 1984-01-01T00:00:00Z
review text: The book under review presents 199 research problems in complex analysis (mostly one variable), especially problems connected with spaces or algebras of analytic functions and with linear operatos. The problems are given by more than 100 mathematicians around the world. This problem book evolved from an earlier version published as ''Studies on linear operators and function theory. 99 unsolved problems of linear and complex analysis'', Zapiski Nauchnykh Seminarov LOMI, Vol. 81 (1978; Zbl 0445.00008). The new book contains all those problems and many more. In the meantime about half of the earlier problems were solved, partly or completely, and the solutions of those problems which are completely solved are given in the last 35 pages of the book. It also contains reports on progress on the solutions of many other problems which has been made from 1978 until 1983. Those informations are very valuable. The problems are divided into 13 groups, each group has its own chapter. Chapter 1 is mainly concerned with Banach space properties of spaces of analytic functions, for example with the problem of classifying the complemented subspaces in the disc algebra \(A({\mathbb{D}})\) or in the Hardy spaces \(H^ 1\) and \(H^{\infty}\). An important problem in this section is the question whether \(H^{\infty}\) has the approximation property (in the sense of Grothendieck) or not. Chapter 2 is devoted to Banach algebras. The most interesting problem here is the corona problem for the algebra \(H^{\infty}(G)\) where G is an arbitrary planar domain. A good reference in case of finitely-connected domains G is the book of \textit{S. Fisher} [Function theory on planar domains (1983; Zbl 0511.30022)]. For infinitely-connected domains some progress is made in a recent paper of \textit{M. Behrens} [Trans. Am. Math. Soc. 286, 203-226 (1984)]. After a short chapter on problems of probabilistic origin (stationary Gaussian processes) chapter 4, the largest in the book, turns to different questi…
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 30-01
Mathematics Subject Classification ID: 46-01
Mathematics Subject Classification ID: 00A07
zbMATH DE Number: 3867692
zbMATH Keywords: Hankel operators
zbMATH Keywords: disc algebra
zbMATH Keywords: corona problem
zbMATH Keywords: Toeplitz operators
zbMATH Keywords: BMO- functions
zbMATH Keywords: Bloch spaces
zbMATH Keywords: vanishing mean oscillation
zbMATH Keywords: prime ideal
zbMATH Keywords: Hoffman's theory of maximal ideals
zbMATH Keywords: interpolating Blaschke products
zbMATH Keywords: Painlevé null sets
zbMATH Keywords: Denjoy conjecture
reviewed by: Michael von Renteln
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0072183
OpenAlex ID: W3088029432
DOI: 10.1007/BFB0072183
description: scientific article; zbMATH DE number 3867692