Linear Operators: Spectral theory |
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Page 1045
... integral ( 1 ) exists for almost all x , and defines a bounded mapping of L , ( E " ) into itself , 1 ≤ p ≤ co . For p and = p 2 , the exact norm of this mapping may be determined . For p = 2 , the n - dimensional analogue of Theorem ...
... integral ( 1 ) exists for almost all x , and defines a bounded mapping of L , ( E " ) into itself , 1 ≤ p ≤ co . For p and = p 2 , the exact norm of this mapping may be determined . For p = 2 , the n - dimensional analogue of Theorem ...
Page 1046
Nelson Dunford, Jacob T. Schwartz. an integral studied by Hilbert . The integral ( 2 ) may be interpreted in terms of a Cauchy principal value as • + 00 pixy Ꮖ dx = lim = 01-3 lim 8 • ∞0 pixy . + ∞ eixy dx Ꮖ ε -ixy - e dx x ∞ sin xy ...
Nelson Dunford, Jacob T. Schwartz. an integral studied by Hilbert . The integral ( 2 ) may be interpreted in terms of a Cauchy principal value as • + 00 pixy Ꮖ dx = lim = 01-3 lim 8 • ∞0 pixy . + ∞ eixy dx Ꮖ ε -ixy - e dx x ∞ sin xy ...
Page 1047
... integral f ( x ) │x - y│ dx instead of ( 3 ) , all our considerations would fail . In the multi - dimensional case the convolution integrals ( 4 ) • + ∞ Q ( x - y ) | xy | n f ( y ) dy of the form analyzed by Calderón and Zygmund ...
... integral f ( x ) │x - y│ dx instead of ( 3 ) , all our considerations would fail . In the multi - dimensional case the convolution integrals ( 4 ) • + ∞ Q ( x - y ) | xy | n f ( y ) dy of the form analyzed by Calderón and Zygmund ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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