Linear and complex analysis. Problem book. 199 research problems
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 questions in operator theory. The problems are connected with selfadjoint spectral theory, including perturbation theory and scattering theory, Banach algebra methods, ''near-normal'' operators, etc. Hankel- and Toeplitz operators have their own chapter, since they play a decisive role in a wide range of problems in complex analysis, e.g. moment problems, orthogonal polynomials, prediction theory, boundary and interpolation problems of analytic functions. - Chapter 6 continues with singular integrals, BMO-functions, Bloch spaces, for example to characterize the Blaschke products which belong to the Bloch space \(B_ 0\) by means of the distribution of their zeros. Another very interesting problem in this section is to determine the structure of the algebra of all \(H^{\infty}\)-functions whose boundary values have vanishing mean oscillation. - The next chapter is devoted to problems in spectral analysis and synthesis. Interesting are the problems connected with localization of ideals in algebras of analytic functions determined by growth conditions. Also questions about weak invertibility (or cyclic vectors) in spaces of analytic functions are considered. Regarding this matter the interested reader should also consult the recent paper of \textit{L. Brown} and \textit{A. L. Shields} [Trans. Am. Math. Soc. 285, 269-304 (1984; Zbl 0537.30040)]. Also problems connected with invariant subspaces are presented, e.g. what are the invariant subspaces of the backward shift operator in the Smirnov class \(N^+\). - Then problems of ideal theory in algebras of analytic functions are presented, e.g. is every nonzero finitely generated prime ideal in \(H^{\infty}\) a maximal ideal of the form \(M(z_ 0)=\{f\in H^{\infty}:\quad f(z_ 0)=0\},\quad z_ 0\in {\mathbb{D}}?\) This question has been answered in the positive by \textit{R. Mortini} [Zur Idealstruktur der Disk-Algebra \(A({\mathbb{D}})\) und der Alebra \(H^{\infty}\) (1984; Zbl 0545.30039)] and independently by \textit{P. Gorkin} (see e.g. Zbl 0545.30040). Both authors used Hoffman's theory of maximal ideals and interpolating Blaschke products. Perhaps the most well known question in ideal theory of \(H^{\infty}\) (and some other algebras of holomorphic functions) is the following: If we have \(| f| \leq C\sum^{n}_{1}| f_ i|\), must there be functions \(g_ 1,...,g_ n\in H^{\infty}\) with \(f^{\alpha}=g_ 1f_ 1+...+g_ nf_ n,\) where \(\alpha =2?\) For \(\alpha =3\) this is true, for \(\alpha =1\) it is false. The case \(\alpha =2\) implies the case \(\alpha =3\) and from this the corona theorem is easily deduced (take \(f=1)\). - The next chapter is devoted to approximation and capacity problems, including classical uniform approximation, best approximation in the sense of Tschebysheff, Padé approximations, metric estimates of capacities, analytic capacity and Painlevé null sets. The last mentioned problem became very interesting in view of Caldeŕons solution of the famous Denjoy conjecture. - Chapter 9 deals with sets of uniqueness and moment problems. The last four chapters present various problems in interpolation, bases, multipliers, entire and subharmonic functions, etc. For every mathematician working in analysis this book is of great interest and value. We highly recommend it.
scientific article; zbMATH DE number 3867692