Linear Operators: Spectral theory |
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Page 926
... weakly , or n n ( b ) ( Txn , xn ) → 0 whenever an →→ 0 weakly . ( Hint : For ( b ) , show that the hypothesis implies that ( Tan , Yn ) → 0 whenever the sequences { x } and { y } are both weakly convergent to zero . ) 9. Notes and ...
... weakly , or n n ( b ) ( Txn , xn ) → 0 whenever an →→ 0 weakly . ( Hint : For ( b ) , show that the hypothesis implies that ( Tan , Yn ) → 0 whenever the sequences { x } and { y } are both weakly convergent to zero . ) 9. Notes and ...
Page 932
... weakly countably additive ) function on to the set of positive operators on a Hilbert space § satis- fying F ( $ ) = 0 and F ( S ) = I. Then there exists a Hilbert space 25 and a self adjoint projection valued additive ( resp . weakly ...
... weakly countably additive ) function on to the set of positive operators on a Hilbert space § satis- fying F ( $ ) = 0 and F ( S ) = I. Then there exists a Hilbert space 25 and a self adjoint projection valued additive ( resp . weakly ...
Page 1776
... weakly complete and a subset is weakly sequentially compact if and only if it is bounded . PROOF . This follows from 6 , II.3.28 , and II.3.29 . Q.E.D. 8 DEFINITION . A set ACH is called an orthonormal 1776 APPENDIX.
... weakly complete and a subset is weakly sequentially compact if and only if it is bounded . PROOF . This follows from 6 , II.3.28 , and II.3.29 . Q.E.D. 8 DEFINITION . A set ACH is called an orthonormal 1776 APPENDIX.
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