Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert SpaceThis classic text, written by two notable mathematicians, constitutes a comprehensive survey of the general theory of linear operations, together with applications to the diverse fields of more classical analysis. Dunford and Schwartz emphasize the significance of the relationships between the abstract theory and its applications. This text has been written for the student as well as for the mathematician—treatment is relatively self-contained. This is a paperback edition of the original work, unabridged, in three volumes. |
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
Exercises | 879 |
Notes and Remarks | 883 |
Bounded Normal Operators in Hilbert Space | 887 |
The Spectral Theorem for Bounded Normal Operators | 895 |
Notes and Remarks | 1145 |
The Spectral Theorem for Unbounded Self Adjoint Operators | 1191 |
Spectral Representation of Unbounded Self Adjoint Trans | 1205 |
The Extensions of a Symmetric Transformation | 1222 |
Semibounded Symmetric Operators | 1241 |
The Canonical Factorization | 1249 |
Exercises | 1259 |
Notes and Remarks | 1271 |
Eigenvalues and Eigenvectors | 902 |
Unitary Self Adjoint and Positive Operators | 905 |
Spectral Representation | 909 |
A Formula for the Spectral Resolution | 920 |
Perturbation Theory | 921 |
Exercises | 923 |
Notes and Remarks | 926 |
Miscellaneous Applications | 937 |
Almost Periodic Functions | 945 |
Convolution Algebras | 949 |
Closure Theorems | 978 |
Exercises | 1001 |
HilbertSchmidt Operators | 1009 |
The Hilbert Transform and the CalderónZygmund Inequality | 1044 |
Exercises | 1073 |
The Classes C of Compact Operators Generalized Carleman Inequalities | 1088 |
Subdiagonalization of Compact Operators | 1119 |
Ordinary Differential Operators | 1278 |
Adjoints and Boundary Values of Differential Operators | 1285 |
Resolvents of Differential Operators | 1311 |
Compact Resolvents | 1331 |
Qualitative Theory of the Deficiency Index | 1393 |
Qualitative Theory of the Spectrum | 1443 |
Examples | 1505 |
Exercises | 1539 |
Notes and Remarks | 1582 |
Algebras of Spectral Operators | 1612 |
Linear Partial Differential Equations and Operators | 1629 |
The Theorem of Sobolev | 1683 |
Some Geometric Considerations | 1701 |
The Elliptic Boundary Value Problem | 1707 |
APPENDIX | 1773 |
Unbounded Spectral Operators | 1789 |
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Common terms and phrases
adjoint extension adjoint operator algebra analytic B-algebra Borel set boundary conditions boundary values bounded operator C₁ C₂ closed closure Co(I coefficients compact operator complex numbers constant continuous function converges Corollary deficiency indices Definition denote dense eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real axis satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₁(t T₂ theory To(t topology unique unitary vanishes vector zero