Theory of Np Spaces: (Frontiers in Mathematics)

This monograph provides a comprehensive study of a typical and novel function space, known as the $\mathcal{N}_p$ spaces. These spaces are Banach and Hilbert spaces of analytic functions on the open unit disk and open unit ball, and the authors also explore composition operators and weighted composition operators on these spaces. The book covers a significant portion of the recent research on these spaces, making it an invaluable resource for those delving into this rapidly developing area. The authors introduce various weighted spaces, including the classical Hardy space $H^2$, Bergman space $B^2$, and Dirichlet space $\mathcal{D}$. By offering generalized definitions for these spaces, readers are equipped to explore further classes of Banach spaces such as Bloch spaces $\mathcal{B}^p$ and Bergman-type spaces $A^p$. Additionally, the authors extend their analysis beyond the open unit disk $\mathbb{D}$ and open unit ball $\mathbb{B}$ by presenting families of entire functions in the complex plane $\mathbb{C}$ and in higher dimensions. The Theory of $\mathcal{N}_p$ Spaces is an ideal resource for researchers and PhD students studying spaces of analytic functions and operators within these spaces.

Table of Contents:
Chapter. 1. Function spaces.- Chapter. 2. The counting function and its applications.- Chapter. 3. Np-spaces in the unit disc D.- Chapter. 4. The alpha-Bloch spaces.- Chapter. 5. Weighted composition operators on D.- Chapter. 6. Hadamard gap series in H.- Chapter. 7. Np spaces in the unit ball B.- Chapter. 8. Weighted composition operators on B.- Chapter. 9. Structure of Np-spaces in the unit ball B.- Chapter. 10. Composition operators between Np and Nq.- Chapter. 11. Np-type functions with Hadamard gaps in the unit ball B.- Chapter. 12. N (p; q; s)-type spaces in the unit ball of Cn.

About the Author :
Javad Mashreghi is an esteemed mathematician and author renowned for his work in the areas of functional analysis, operator theory, and complex analysis. He has made significant contributions to the study of analytic function spaces and the operators that act upon them. Prof. Mashreghi has held various prestigious positions throughout his career. He served as the 35th President of the Canadian Mathematical Society (CMS) and has been recognized as a Lifetime Fellow of both CMS and the Fields Institute. He currently holds the Canada Research Chair at Université Laval and has also been honored as a Fulbright Research Chair at Vanderbilt University. Le Hai Khoi is an expert in the fields of function spaces and operator theory, with a particular focus on the representation of functions using series expansions involving exponential functions, rational functions, and Dirichlet series. He has made significant contributions to these areas and has a prolific research output, having published over 80 research papers in the relevant field. Prof. Le Hai Khoi is well-known for his expertise and active involvement in the study of $\mathcal{N}_p$ spaces, which are Banach and Hilbert spaces of analytic functions on the open unit disk and open unit ball.

Review :
"The book is well written and easy to read. Every chapter ends with a short note which gives references for the results. The book provides a useful source for young researchers to gain basic knowledge and various techniques to initiate their own research in the field of operator theory and the theory of spaces of analytic functions." (Ruhan Zhao, Mathematical Reviews, February, 2025)  “The book is pedagogical and detailed. Each chapter ends with a ‘Notes’ section, putting much of the material in historical context, while providing useful additional references. The book is a valuable guide to researchers interested in the topic.” (Amol Sasane, zbMATH 1539.46001, 2024)