Perturbation Theory for Linear OperatorsSpringer Science & Business Media, 6 déc. 2012 - 623 pages In view of recent development in perturbation theory, supplementary notes and a supplementary bibliography are added at the end of the new edition. Little change has been made in the text except that the para graphs V-§ 4.5, VI-§ 4.3, and VIII-§ 1.4 have been completely rewritten, and a number of minor errors, mostly typographical, have been corrected. The author would like to thank many readers who brought the errors to his attention. Due to these changes, some theorems, lemmas, and formulas of the first edition are missing from the new edition while new ones are added. The new ones have numbers different from those attached to the old ones which they may have replaced. Despite considerable expansion, the bibliography i" not intended to be complete. Berkeley, April 1976 TosIO RATO Preface to the First Edition This book is intended to give a systematic presentation of perturba tion theory for linear operators. It is hoped that the book will be useful to students as well as to mature scientists, both in mathematics and in the physical sciences. |
Table des matières
| 1 | |
| 10 | |
| 16 | |
The adjoint operator | 23 |
Operators in unitary spaces | 47 |
Pairs of projections | 56 |
Chapter | 62 |
Perturbation series | 74 |
The spectral theorem and perturbation of spectral families | 353 |
Chapter Seven | 364 |
Holomorphic families of type A | 375 |
Selfadjoint holomorphic families | 385 |
Holomorphic families of type B | 393 |
Further problems of analytic perturbation theory | 413 |
Chapter Eight | 426 |
Asymptotic expansions | 441 |
The eigenvalue problem | 75 |
Convergence radii and error estimates | 88 |
Similarity transformations of the eigenspaces and eigenvectors | 98 |
Perturbation of symmetric operators | 120 |
Chapter Three | 126 |
Linear operators in Banach spaces | 142 |
Bounded operators | 149 |
Compact operators | 157 |
Closed operators | 163 |
Resolvents and spectra | 172 |
Chapter Four | 189 |
Generalized convergence of closed operators | 197 |
Perturbation of the spectrum | 208 |
Pairs of closed linear manifolds | 218 |
Stability theorems for semiFredholm operators | 229 |
Degenerate perturbations | 244 |
Hilbert space | 251 |
Unbounded operators in Hilbert spaces | 267 |
Perturbation of selfadjoint operators | 287 |
operators | 293 |
Chapter | 308 |
The representation theorems | 322 |
Perturbation of sesquilinear forms and the associated operators | 336 |
Quadratic forms and the Schrödinger operators | 343 |
34 | 450 |
Generalized strong convergence of sectorial operators | 453 |
10 | 456 |
Asymptotic expansions for sectorial operators | 463 |
Spectral concentration | 473 |
Chapter Nine | 479 |
Applications to the heat and Schrödinger equations | 495 |
Perturbation of semigroups | 497 |
Approximation by discrete semigroups | 509 |
Chapter | 516 |
The trace and determinant | 525 |
Existence and completeness of wave operators | 537 |
A stationary method | 553 |
Solution of the integral equation for rank | 560 |
Chapter I | 568 |
Chapter VI | 573 |
509 | 584 |
Supplementary Bibliography | 596 |
| 607 | |
| 612 | |
| 613 | |
| 619 | |
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Expressions et termes fréquents
adjoint analytic assume Banach space belongs boundary condition bounded operator bounded-holomorphic Cauchy sequence closable closed linear manifold coefficients commutes compact resolvent complete consider continuous convergence denote densely defined differential operator dist domain easily seen eigen eigenprojections eigenvalues eigenvalues of T(x eigenvectors equation Example exists finite finite-dimensional follows Friedrichs extension function given H₁ H₂ Hence Hilbert space holomorphic family implies inequality integral operator inverse isolated eigenvalue Lemma linear operator m-sectorial M₂ matrix multiplicity nonnegative norm orthogonal projection P₁ P₂ perturbation theory Problem proof of Theorem proved PT(¹ R₂ relatively bounded Remark replaced restriction satisfied selfadjoint selfadjoint operator semigroup sense sequence sesquilinear form space H spectral spectrum subset subspace sufficiently symmetric operator T-bounded T₁ T₂ true u₁ unitary unitary space vector zero