Linear Operators, Part 2 |
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Page 1246
... dense subspace into the space 1. In this case A is still continuous , for D ( T ) of | Ax | } = ( Ax , Ax ) , ( Ax ... dense subset of itself , we see B is an operator in § of norm at most one . Since for x and y in the dense set D ( T ) ...
... dense subspace into the space 1. In this case A is still continuous , for D ( T ) of | Ax | } = ( Ax , Ax ) , ( Ax ... dense subset of itself , we see B is an operator in § of norm at most one . Since for x and y in the dense set D ( T ) ...
Page 1271
... dense in H. Then if x is in D ( T ) , we have | ( T + iI ) x | 2 = ( Tx , Tx ) ‡ i ( x , Tx ) ± i ( Tx , x ) + ( x , x ) = = Tx2 + x2x2 . This shows that if ( T + iI ) x 0 , then x = O and so the operators Til have inverses . Let V be ...
... dense in H. Then if x is in D ( T ) , we have | ( T + iI ) x | 2 = ( Tx , Tx ) ‡ i ( x , Tx ) ± i ( Tx , x ) + ( x , x ) = = Tx2 + x2x2 . This shows that if ( T + iI ) x 0 , then x = O and so the operators Til have inverses . Let V be ...
Page 1905
... Dense convex sets , V.7.27 ( 437 ) Dense linear manifolds , V.7.40-41 ( 438-439 ) Dense set , definition , I.6.11 ( 21 ) density of continuous functions in TM and L ,, III.9.17 ( 170 ) , IV.8.19 ( 298 ) density of simple functions in L ...
... Dense convex sets , V.7.27 ( 437 ) Dense linear manifolds , V.7.40-41 ( 438-439 ) Dense set , definition , I.6.11 ( 21 ) density of continuous functions in TM and L ,, III.9.17 ( 170 ) , IV.8.19 ( 298 ) density of simple functions in L ...
Contents
BAlgebras | 861 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure 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 real numbers satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology unique unitary vanishes vector zero