This self-contained encyclopedic monograph gives a detailed introduction to Bézout equations and stable ranks, encompassing and explaining needed topological, analytical, and algebraic tools and methods. Some of the highlights included are Carleson's corona theorem and the Bass, topological, and matricial stable ranks. The first volume focusses on topological structures, Banach algebras, and advanced function theory, thus preparing the stage for the algebraic structures in the second volume towards examining stable ranks with analytic methods. The main emphasis is laid on algebras of holomorphic functions. Often a new approach is presented or at least a different angle of sight, which makes the book attractive both for researchers and students interested in these active fields of research.
This self-contained encyclopedic monograph gives a detailed introduction to Bézout equations and stable ranks, encompassing and explaining needed topological, analytical, and algebraic tools and methods. Some of the highlights included are Carleson's corona theorem and the Bass, topological, and matricial stable ranks. The first volume focusses on topological structures, Banach algebras, and advanced function theory, thus preparing the stage for the algebraic structures in the second volume towards examining stable ranks with analytic methods. The main emphasis is laid on algebras of holomorphic functions. Often a new approach is presented or at least a different angle of sight, which makes the book attractive both for researchers and students interested in these active fields of research.
The classical ℓp sequence spaces have been a mainstay in Banach spaces. This book reviews some of the foundational results in this area (the basic inequalities, duality, convexity, geometry) as well as connects them to the function theory (boundary growth conditions, zero sets, extremal functions, multipliers, operator theory) of the associated spaces ℓpA of analytic functions whose Taylor coefficients belong to ℓp. Relations between the Banach space ℓp and its associated function space are uncovered using tools from Banach space geometry, including Birkhoff-James orthogonality and the resulting Pythagorean inequalities. The authors survey the literature on all of this material, including a discussion of the multipliers of ℓpA and a discussion of the Wiener algebra ℓ1A. Except for some basic measure theory, functional analysis, and complex analysis, which the reader is expected to know, the material in this book is self-contained and detailed proofs of nearly all the results are given. Each chapter concludes with some end notes that give proper references, historical background, and avenues for further exploration.
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