Algebraic analysis. (English) Zbl 0696.47002

What is “Algebraic Analysis”? The name “Algebraic Analysis” was used by Lagrange in a subtitle to the second revised and enlarged edition of his “Théorie des fonctions analytiques” (1813). The same subtitle was used by Cauchy in 1821 in his “Cours d’analyse de l’École Royale Politechnique, 1re partie, Analyse algébrique”. In his introduction he wrote “As to methods, I have sought to make them as rigorous as those of geometry, so as never to have recourse to justifications drawn from the generality of algebra”.
The term “algebraic analysis” appears in the title of over a dozen books without a clear delineation of what it describes; often it is used in contexts where the common thread is tenuous or doesn’t exist. Of the older books we mention “Instituzioni di Analysi Algebrica” by A. Capelli (Napoli, 1894); “Corso di Analysi Algebrica con Introduzione al Calcolo Infinitesimale” by E. Cesàro (Torino, 1894); “Elementares Lehrbuch der Algebraischen Analysis und der Infinitesimal Rechnung”, also by E. Cesàro (Leipzig, 1904), “Course of Algebraic Analysis” (in Russian, Kiev, 1911) and “Treatise on Algebraic Analysis” (in Russian and Ukrainian, Kiev, 1938-1939; Zbl 0020.19705 and Zbl 0020.19706) by D. O. Grave. Capelli’s book concerns algebraic curves, the two books by Grave are devoted to algebraic problems, while Cesàro’s book is an attempt to a common treatment of Algebra, Linear Algebra, Calculus and Differential Equations, close to what is often called nowadays “linear analysis”.
In 1988 two volumes entitled “Algebraic Analysis” (Vol I: Zbl 0665.00008), were published. Edited by M. Kashiwara and T. Kawai, the two volumes consist of papers dedicated to Professor Mikio Sato, “the initiator of algebraic analysis in the twentieth century”, whose research seems to aim at the renaissance of “Algebraic Analysis” of Euler, and deals with the theory of hyperfunctions (which Sato invented in 1957) and with other topics not related to the classical books mentioned earlier. Finally, we mention “Foundations of Algebraic Analysis”, Princeton (1986; Zbl 0605.35001) by M. Kashiwara, T. Kawai and T. Kimura which is concerned with microlocal analysis.
The author of the book under review has her own very interesting explanation of what led to the type of “Algebraic Analysis” considered in her book. But it is clear from above that “Algebraic Analysis” means markedly different things to different authors; one has to examine the meaning from the context in which it is used. For the present book, this is best highlighted by quoting titles of the main chapters and key phrases: Calculus of right invertible operators, general solution of equations with right invertible operators, initial and boundary value problems, well-posed and ill-posed boundary value problems, periodic operators and elements, shift operators and shift invariant subspaces, D- algebras, perturbations and nonlinear problems, metric properties in algebraic analysis.
The common thread and concepts throughout the book (9 chapters) are the proper definition of initial operators for right invertible operators and their fundamental properties, and “Calculus in Algebraic Analysis” by which the author means the theory of right invertible operators in linear spaces (without any topology, in general) - think of indefinite integrals!
Reviewer: M.Z.Nashed


47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
47A50 Equations and inequalities involving linear operators, with vector unknowns