Introduchon Chapter One Operator theory in finite—dimensional vector spaces §1.Vector spaces and normed vector spaces 1.Basic nohons 2.Bases 3.Linear manifolds 4.Convergence and norms 5.Topological nohons in a normed space 6.Infinite series of vectors 7.Vector—valued funchons §2.Linear forms and the adjoint space 1.Linear forms 2.The adjoint space 3.The adjoint basis 4.The adjoint space of a normed space 5.The convexity of balls 6.The second adjoint space §3.Linear operators 1.Definihons.Matrix representations 2.Linear operations on operators 3.The algebra of linear operators 4.Projections.Nilpotents 5.Invariance.Decomposihon 6.The adjoint operator §4.Analysis with operators 1.Convergence and norms for operators 2.The norm of T 3.Examples of norms 4.Infinte series of operators 5.Operator—valued functions 6.Pairs of projechons §5.The eigenvalue problem 1.Definihons 2.The resolvent 3.Singularities of the resolvent 4.The canorucal form of an operator 5.The adjoint problem 6.Functions of an operator 7.Similarity transformations §6.Operators in unitary spaces 1.Unitary spaces 2.The adjoint space 3.Orthonormal families 4.Linear operators 5.Symmetric forms and symmetric operators 6.Unitary，isometric and normal operators 7.Projections 8.Pairs of projections 9.The eigenvalue problem 10.The minimax principle
Chapter Two Perturbatlon theory in a finite—dimensional space §1.Analyhc perturbahon of eigenvalues 1.The problem 2.Singularities of the eigenvalues 3.Perturbation of the resolvent 4.Perturbation of the eigenprojections 5.Singularities of the eigertprojections 6.Remarks and examples 7.The case of T（x） linear in x 8.Summary §2.Perturbation series 1.The total projection for the A—group 2.The weighted mean of eigenvalues 3.The reduction process 4.Formulas for higher approximahons 5.A theorem of MOTZKIN—TAUSSKY 6.The ranks of the coefficients of the perturbation series §3.Convergence radu and error estimates 1.Simple estimates 2.The method of majorizing series 3.Estimates on eigenvectors 4.Further error eshmates 5.The special case of a normal unperturbed operator 6.The enumerahve method §4.Similarity transformations of the eigenspaces and eigenvectors 1.Eigenvectors 2.Transformation funchons 3.Soluhon of the dffierential equahon 4.The transformation function and the reduction process 5.Simultaneous transformahon for several projections 6.Diagonalization of a holomorphic matrix function §5.Non—analytic perturbations 1.Continuity of the eigenvalues and the total projechon 2.The numbering of the eigenvalues 3.Conhnuity of the eigenspaces and eigenvectors 4.Differentiability at a point 5.Differenhability in an interval …… Chapter Three Introduction to the theory of operators in Banach spaces Chapter Four Stability theorems Chapter Five Operators in Hilbert spaces Chapter Six Sesquilinear forms in Hilbert spaces and associated operators Chapter Seven Analytic perturbation theory Chapter Eight Asymptotic perturbation theory Chapter Nine Perturbation theory for semigroups of operators Chapter Ten Perturbauuon of continuous spectra and unitary equivalence Supplementary Notes
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