Analytic perturbation theory for matrices and operators. Licensed ed
This valuable monograph is an extended and partly modified English version of the author's book ``Endlichdimensionale analytische Störungstheorie.'' Berlin: Akademie-Verlag (1972; Zbl 0303.47012). Both the original version and the present book present essential supplements to \textit{T. Kato}'s fundamental monograph ``Perturbation theory for linear operators.'' Berlin etc.: Springer(1966; Zbl 0148.12601). In his book the author gves a systematic presentation of analytic perturbation theory for matrices and the implications of this theory for analytic perturbations of isolated eigenvalues of finite algebraic multiplicity of closed linear operators in Banach spaces. The central topic of the book is a complete description of how the structure of a matrix is changed by analytic perturbation. The main results can be described as follows: Let \(A(\cdot)\) be a matrix-valued function, defined in a subset \(G\) of the complex plane. Let \(\xi\in G\). Denote the eigenvalues of \(A(\xi)\) by \(\lambda_i(\xi)\) and the corresponding eigenprojections by \(P_i(\xi)\), \[ P_i(\xi) := \frac{1}{2\pi i}\int_{\Gamma_i}(\lambda - A(\xi))^{-1}\,d\lambda \quad (i=1,2,\ldots,r(\xi)); \] each path \(\Gamma_i\) of integration is a positively oriented circle with center \(\lambda_i(\xi)\) which separates \(\lambda_i(\xi)\) from the remaining spectrum of \(A(\xi)\). The operator \[ D(\xi) := A(\xi)- \sum^{r(\xi)}_{i=1}\lambda_i(\xi)P_i(\xi) \] is nilpotent. The function \(A(\cdot)\) is said to have an invariant Jordan structure in \(G_0\subset G\) if \(r(\xi)\) is constant in \(G_0\), the dimensions of the ranges of the projections \(P_i(\xi)\) are constant in \(G_0\) (for a suitable numbering) and there is a constant nilpotent matrix \(N\) such that \(D(\xi)\) is similar to \(N\) for each \(\xi \in G_0\). Theorem: Let \(G\) be a domain and \(A(\cdot)\) be holomorphic (meromorphic) in \(G\). Then there is an at most countable set \(A\subset G\), having no accumulation points in \(G\), such that A(.) has an invariant Jordan structure in \(G\setminus A\). Corollary: Let \(A(\cdot)\) be a holomorphic perturbation of \(A_0=A(0)\) in \(| z| <R\), then there is a \(\rho\in (0,R]\) such that \(A(\cdot)\) has an invariant Jordan structure in \(0<| z| <\rho\). This corollary is closely related to the following facts: All perturbed spectral quantities can be expanded into Puiseux series. The expansions of eigenvalues, eigenvectors and Jordan vectors are power series of the type \[ a_0+\sum^{\infty}_{\nu =1}a_{\nu}z^{\nu /p}\quad (| z| <\rho); \] the expansions of eigenprojections and irreducible projections are Laurent series of the form \[ \sum^{\infty}_{\nu =-N}P_{\nu}z^{\nu /p}\quad (0<| z| <\rho); \] here the \(p\) are certain (finitely many) natural numbers and the \(a_0\) are the unperturbed spectral objects (eigenvalue, eigenvector, Jordan vector). Similar statements hold for analytic perturbations of closed linear operators in Banach spaces with respect to isolated eigenvalues of finite algebraic multiplicity. The main part of the book (as the original German version) is devoted to the analytic perturbation theory for one complex variable. In the present book a complete chapter deals with the transfer of the essential results to the case of analytic perturbations depending on several complex variables. This supplement to the original version is based on the author's recent paper on ``Analytic perturbation theory for linear operators depending on several complex variables'' [Mat. Issled. 9, No. 1(31), 17--39 (Russian) (1974; Zbl 0306.47008)]. Throughout all parts of the book well chosen examples illustrate the achieved results. The book is self-contained since well composed reviews of the necessary material from algebra, operator theory and complex analysis are presented. Chapter headings: 1. Finite dimensional linear analysis (A. Basic algebraic concepts, B. Analysis on finite-dimensional complex linear spaces), 2. Spectral theory and Jordan structure of linear operators, 3. Spectral theory of meromorphic operator-valued functions, 4. Linear operators on linear spaces over commutative fields; first applications, 5. Jordan structure of meromorphic operator-valued functions, 6. Analytic perturbations, 7. Reduction theory, 8. Numerical analysis for perturbation series - convergence radii and error estimates, 10. Analytic perturbations in Banach spaces, Supplement. Analytic perturbations for linear operators depending on several complex variables, Appendix. Basic material from complex function theory and algebra.
analytic perturbation theory for matrices
analytic perturbations of isolated eigenvalues of finite algebraic
multiplicity of closed linear operators in Banach spaces
expansions of eigenprojections and irreducible projections
analytic perturbations of isolated eigenvalues
finite algebraic multiplicity of closed linear operators
scientific article; zbMATH DE number 3949073