Linear Operators: Spectral theory |
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Page 1224
... symmetric then every symmetric extension T2 of T1 , and , in particular , every self adjoint extension of T1 , satisfies T1CT2 CTC T * . 1 2 2 PROOF . If T1CT2 and y = D ( T * ) , then ( x , T * y ) = ( T2x , y ) ΞΙ 2 ( T1x , y ) for ...
... symmetric then every symmetric extension T2 of T1 , and , in particular , every self adjoint extension of T1 , satisfies T1CT2 CTC T * . 1 2 2 PROOF . If T1CT2 and y = D ( T * ) , then ( x , T * y ) = ( T2x , y ) ΞΙ 2 ( T1x , y ) for ...
Page 1236
... symmetric family of boundary conditions , B ( x ) = 0 , i = 1 , . . . , k . Conversely , every such restriction T1 of T * is a closed symmetric extension of T. 1 PROOF . We shall prove the second statement first . As each B¿ is a ...
... symmetric family of boundary conditions , B ( x ) = 0 , i = 1 , . . . , k . Conversely , every such restriction T1 of T * is a closed symmetric extension of T. 1 PROOF . We shall prove the second statement first . As each B¿ is a ...
Page 1238
... symmetric operator with finite deficiency indices whose sum is p . Let A1 , ... , A , be a complete set of boundary values for T , and let Σ3 ̧¡_1¤¿¡Â¿Ã ̧ be the bilinear form of Lemma 23 . A set of boundary conditions 1P , 4¿ ( x ) = 0 ...
... symmetric operator with finite deficiency indices whose sum is p . Let A1 , ... , A , be a complete set of boundary values for T , and let Σ3 ̧¡_1¤¿¡Â¿Ã ̧ be the bilinear form of Lemma 23 . A set of boundary conditions 1P , 4¿ ( x ) = 0 ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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(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 satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology tr(T unique unitary vanishes vector zero