Linear Operators: Spectral theory |
From inside the book
Results 1-3 of 74
Page 1675
... present proof , F , F , so that a , F is in H ( C ) . This completes the proof of the direct part of ( i ) of the present lemma . - To prove the converse , let F be in H ( C ) and let 1 be in H ( C ) . Let us agree to consider that each ...
... present proof , F , F , so that a , F is in H ( C ) . This completes the proof of the direct part of ( i ) of the present lemma . - To prove the converse , let F be in H ( C ) and let 1 be in H ( C ) . Let us agree to consider that each ...
Page 1679
... present lemma it suffices to show that ( 4 ) G ( S ̧ ¥ ( • ) q ( • − y ) f ( y ) dy ) = { , G ( v ( • ) q ( • − y ) ) f ( y ) dy , where G = F. Let K2 be a compact subset of I containing in its interior a second compact set outside of ...
... present lemma it suffices to show that ( 4 ) G ( S ̧ ¥ ( • ) q ( • − y ) f ( y ) dy ) = { , G ( v ( • ) q ( • − y ) ) f ( y ) dy , where G = F. Let K2 be a compact subset of I containing in its interior a second compact set outside of ...
Page 1692
... present lemma . Q.E.D. 9 COROLLARY . The conclusions of Corollary 6 and Lemma 8 remain valid even if the open set I of these results is replaced by the cube C = { x E " | x | < л , j = 1 , ... , n } . PROOF . It was observed , in the ...
... present lemma . Q.E.D. 9 COROLLARY . The conclusions of Corollary 6 and Lemma 8 remain valid even if the open set I of these results is replaced by the cube C = { x E " | x | < л , j = 1 , ... , n } . PROOF . It was observed , in the ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Copyright | |
59 other sections not shown
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