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Function Theory and Lp Spaces

Raymond Cheng, Javad Mashreghi, and William T. Ross
Publication Date: 
Number of Pages: 
University Lecture Series
[Reviewed by
Frédéric Morneau-Guérin
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This book addresses two types of spaces that are strongly linked. They are, on the one hand, the \( l^{p} \) spaces with \( 0 < p \leq \infty \), namely the spaces consisting of the \( p \)-power summable sequences, and, on the other, the \( \ell_{A}^{p} \) spaces
\( \ell^{p}_{A} = \left\{ f: \mathbb{D} \rightarrow \mathbb{C} \; | \; f(z)=\sum_{n=0}^{\infty} a_{n}z^{n}, \; (a_{n})_{n=0}^{\infty} \in \ell^{p} \right\} \)
with \( 0 < p \leq \infty \), that is to say, the spaces of the analytic functions defined on the open unit disk and whose sequence of Taylor coefficients belongs to \( \ell^{p} \).
Although the central theme of the book (and this is where its originality lies) is the study of the analytic behavior of functions belonging to the \( \ell_{A}^{p} \) spaces, it is undoubtedly useful to first review the main results of the functional analysis related to the \( \ell^{p} \) spaces. This is what the authors propose to us in the first three chapters before undertaking, in Chapter 4, an exhaustive study of the weak parallelogram laws in the context of the \( \ell^{p} \) spaces for \( 1 < p < \infty \). In Chapter 5, there is a transition from sequence spaces to function spaces and an overview of the theories of Hardy and Bergman spaces.  It is in Chapter 6 that the study of \( \ell^{p}_{A} \) spaces really begins. They are first defined, and then their main properties are presented. In Chapter 7, various operators on \( \ell^{p}_{A} \) (such as the shift operator, the difference quotient operator, and composition operators) are discussed. The study of the shift and the backward shift on \( \ell^{p}_{A} \) is continued in more detail in Chapters 10 and 11. Finally, Chapter 12 deals with the multiplicators of \( \ell^{p}_{A} \), and the thirteenth and last chapter presents an overview of the knowledge relating to \( \ell^{1}_{A} \) space, namely the Wiener algebra.
This book was designed to be accessible to a wide audience. It presupposes only an elementary knowledge of measure theory, functional analysis, and complex analysis. The authors have adopted a simple and clear style, and the evidence is both detailed and patiently put into context historically and mathematically.
Although it contains many results that stand out for their depth, power, and elegance, this book is neither a compendium nor a definitive work on the subject of \( \ell^{p}_{A} \) spaces. How could it be otherwise, given that the theory is still far from being completed? Although they are characterized by the condition of great simplicity, \( \ell^{p}_{A} \) spaces are not as easily understood as the Bergman, Dirichlet, and Hardy spaces. Indeed, the \( p \)-summability condition on the Taylor coefficients of functions belonging to \( \ell^{p}_{A} \) indeed appears to give rise to significantly less regularity. Therefore, many fundamental questions concerning the structure of \( \ell^{p}_{A} \) spaces remain partially or totally unresolved.
There is considerable literature on \( \ell^{p}_{A} \) spaces, though scattered. The authors do undeniably good work in helping to bring it together, giving it structure and organization, and showcasing it. In this book, they establish the necessary basis for the interested reader who hopes to contribute to completing the picture.  
Frédéric Morneau-Guérin is a professor in the Department of Education at Universite TELUQ. He holds a Ph.D. in abstract harmonic analysis.

See the publisher's website.