A Short Introduction to Perturbation Theory for Linear Operators

One Operator theory in finite-dimensional vector spaces.- 1. Vector spaces and normed vector spaces.- 1. Basic notions.- 2. Bases.- 3. Linear manifolds.- 4. Convergence and norms.- 5. Topological notions in a normed space.- 6. Infinite series of vectors.- 7. Vector-valued functions.- 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. Definitions. Matrix representations.- 2. Linear operations on operators.- 3. The algebra of linear operators.- 4. Projections. Nilpotents.- 5. Invariance. Decomposition.- 6. The adjoint operator.- 4. Analysis with operators.- 1. Convergence and norms for operators.- 2. The norm of Tn.- 3. Examples of norms.- 4. Infinite series of operators.- 5. Operator-valued functions.- 6. Pairs of projections.- 7. Product formulas.- 5. The eigenvalue problem.- 1. Definitions.- 2. The resolvent.- 3. Singularities of the resolvent.- 4. The canonical 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.- 11. Dissipative operators and contraction semigroups.- 7. Positive matrices.- 1. Definitions and notation.- 2. The spectral properties of nonnegative matrices.- 3. Semigroups of nonnegative operators.- 4. Irreducible matrices.- 5. Positivity and dissipativity.- Two Perturbation theory in a finite-dimensional space.- 1. Analytic perturbation of eigenvalues.- 1. The problem.- 2. Singularities of the eigenvalues.- 3. Perturbation of the resolvent.- 4. Perturbation of the eigenprojections and eigennilpotents.- 5. Singularities of the eigenprojections.- 6. Remarks and examples.- 7. The case of T(x) linear in x.- 8. Summary.- 2. Perturbation series.- 1. The total projection for the ?-group.- 2. The weighted mean of eigenvalues.- 3. The reduction process.- 4. Formulas for higher approximations.- 5. A theorem of MOTZKIN-TAUSSKY.- 6. The ranks of the coefficients of the perturbation series.- 3. Convergence radii and error estimates.- 1. Simple estimates.- 2. The method of majorizing series.- 3. Estimates on eigenvectors.- 4. Further error estimates.- 5. The special case of a normal unperturbed operator.- 6. The enumerative method.- 4. Similarity transformations of the eigenspaces and eigenvectors.- 1. Eigenvectors.- 2. Transformation functions.- 3. Solution of the differential equation.- 4. The transformation function and the reduction process.- 5. Simultaneous transformation for several projections.- 6. Diagonalization of a holomorphic matrix function.- 7. Geometric eigenspaces (eigenprojections).- 8. Proof of Theorems.8, 4.9 120.- 9. Remarks on projection families and transformation functions.- 5. Non-analytic perturbations.- 1. Continuity of the eigenvalues and the total projection.- 2. The numbering of the eigenvalues.- 3. Continuity of the eigenspaces and eigenvectors.- 4. Differentiability at a point.- 5. Differentiability in an interval.- 6. Asymptotic expansion of the eigenvalues and eigenvectors.- 7. Operators depending on several parameters.- 8. The eigenvalues as functions of the operator.- 6. Perturbation of symmetric operators.- 1. Analytic perturbation of symmetric operators.- 2. Orthonormal families of eigenvectors.- 3. Continuity and differentiability.- 4. The eigenvalues as functions of the symmetric operator.- 5. Applications. A theorem of LIDSKII.- 6. Nonsymmetric perturbation of symmetric operators.- 7. Perturbation of (essentially) nonnegative matrices.- 1. Monotonicity of the principal eigenvalue.- 2. Convexity of the principal eigenvalue.- Notation index.- Author index.