Linear Operators: Spectral theory |
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Page 853
... Section IV.4 , as an appended section on Hilbert space immediately following Chapter XIV . This appended section gives basic defi- nitions and the geometric properties of Hilbert space which are used repeatedly in this volume . Thus ...
... Section IV.4 , as an appended section on Hilbert space immediately following Chapter XIV . This appended section gives basic defi- nitions and the geometric properties of Hilbert space which are used repeatedly in this volume . Thus ...
Page 1392
... Section 8 below . In that section we shall first develop a part of the theory of " special functions , ” and on the basis of this theory , will discuss a number of famous complete orthonormal sets , unitary integral transformations ...
... Section 8 below . In that section we shall first develop a part of the theory of " special functions , ” and on the basis of this theory , will discuss a number of famous complete orthonormal sets , unitary integral transformations ...
Page 1590
... Section 5. The historical development of the main theorems in this section has been sketched in the first section of these notes . We recall that alternate proofs of these results are due to Weyl [ 5 ] ( for the operator of second order ...
... Section 5. The historical development of the main theorems in this section has been sketched in the first section of these notes . We recall that alternate proofs of these results are due to Weyl [ 5 ] ( for the operator of second order ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 closed closure coefficients compact operator complex numbers 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 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 PROOF prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T unique unitary vanishes vector zero