Linear Differential Operators Naimark.pdf
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Linear Differential Operators: A Review of Naimark's Book
Linear differential operators are mathematical objects that act on functions by taking their derivatives. They play an important role in many areas of mathematics and physics, such as differential equations, functional analysis, harmonic analysis, quantum mechanics, and more. In this article, we will review the book Linear Differential Operators by M.A. Naimark, which is a classic and comprehensive reference on the subject.
Naimark's book consists of two parts. The first part covers the elementary theory of linear differential operators, including their definitions, properties, classification, and applications to ordinary differential equations. The second part deals with linear differential operators in Hilbert space, which is a generalization of Euclidean space where functions can be treated as vectors. In this part, Naimark introduces the concepts of self-adjointness, spectrum, eigenvalues, eigenvectors, Green's functions, and boundary value problems for linear differential operators in Hilbert space.
The book is written in a rigorous and clear style, with many examples and exercises to illustrate the theory. It also contains historical notes and references to original sources. The book is suitable for advanced undergraduate and graduate students who have some background in calculus, linear algebra, and complex analysis. It can also serve as a valuable reference for researchers and teachers who work with linear differential operators.
The book was originally published in Russian in 1967 and translated into English by E.R. Dawson in 1968. The English translation was edited by W.N. Everitt and published by F. Ungar Pub. Co. The book is available online as a PDF file[^1^] [^2^].
One of the main topics in the book is the theory of self-adjoint operators, which are linear differential operators that satisfy a certain symmetry condition. Self-adjoint operators have many nice properties, such as being diagonalizable, having real eigenvalues, and having a complete set of orthogonal eigenvectors. Self-adjoint operators also have a close connection with the variational principle, which states that the minimum or maximum value of a certain functional can be found by solving an associated eigenvalue problem.
Another important topic in the book is the theory of boundary value problems, which are problems where the solution of a differential equation is required to satisfy some conditions at the boundary of the domain. Boundary value problems arise naturally in many physical situations, such as heat conduction, wave propagation, electrostatics, and fluid dynamics. Naimark discusses various types of boundary conditions, such as Dirichlet, Neumann, Robin, and periodic, and shows how to use the methods of self-adjoint operators and Green's functions to solve them.
The book also covers some advanced topics, such as the spectral theory of compact operators, the Fredholm alternative, the Sturm-Liouville theory, and the theory of singular differential operators. These topics are useful for studying more general and complex problems involving linear differential operators. Naimark also gives some applications of linear differential operators to partial differential equations, integral equations, and special functions. 061ffe29dd