A short introduction to perturbation theory for linear operators

Tosio Kato


  • One Operator theory in finite-dimensional vector spaces.- x 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.- x 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.- x 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.- x 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.- x 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.- x 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.- x 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.- x 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.- x 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.- x 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.- x 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.- x 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.- x 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.- x 7. Perturbation of (essentially) nonnegative matrices.- 1. Monotonicity of the principal eigenvalue.- 2. Convexity of the principal eigenvalue.- Notation index.- Author index.

「Nielsen BookData」より


書名 A short introduction to perturbation theory for linear operators
著作者等 加藤 敏夫
Kato Tosio
出版元 Springer-Verlag
刊行年月 c1982
版表示 Softcover reprint of the original 1st ed. 1982
ページ数 xiii, 161 p.
大きさ 24 cm
ISBN 0387906665
NCID BA00223053
※クリックでCiNii Booksを表示
言語 英語
出版国 アメリカ合衆国

Clip to Evernote