Linear Operators: Spectral operators |
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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. PROOF . We shall prove the second statement first . As each B¡ is a continuous ...
... 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. PROOF . We shall prove the second statement first . As each B¡ is a continuous ...
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 Σ - 1 , AA , be the bilinear form of Lemma 23 . A set of boundary conditions -1B , 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 Σ - 1 , AA , be the bilinear form of Lemma 23 . A set of boundary conditions -1B , 4 , ( x ) = 0 ...
Page 1272
... symmetric operators . If T is a symmetric operator with dense domain , then it has proper symmetric extensions provided both of its deficiency indices are different from zero . A maximal symmetric operator is one which has no proper ...
... symmetric operators . If T is a symmetric operator with dense domain , then it has proper symmetric extensions provided both of its deficiency indices are different from zero . A maximal symmetric operator is one which has no proper ...
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 Haar measure 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 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 unique unitary vanishes vector zero