Linear Operators: Spectral theory |
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Page 1226
... Lemma 5 ( c ) . Q.E.D. It follows from Lemma 6 ( b ) that any symmetric operator with dense domain has a unique minimal closed symmetric extension . This fact leads us to make the following definition . 7 DEFINITION . The minimal closed ...
... Lemma 5 ( c ) . Q.E.D. It follows from Lemma 6 ( b ) that any symmetric operator with dense domain has a unique minimal closed symmetric extension . This fact leads us to make the following definition . 7 DEFINITION . The minimal closed ...
Page 1698
... Lemma 2.1 , let y be a function in Co ° ( V ) with y ( x ) 1 for x in K. Then , by Lemmas 3.22 and 3.10 , ƒo1 = y ... lemma follows immediately from Lemma 3.48 . Next consider case ( b ) . Using Lemma 3.12 , extend fo¶Ã1 to an element F ...
... Lemma 2.1 , let y be a function in Co ° ( V ) with y ( x ) 1 for x in K. Then , by Lemmas 3.22 and 3.10 , ƒo1 = y ... lemma follows immediately from Lemma 3.48 . Next consider case ( b ) . Using Lemma 3.12 , extend fo¶Ã1 to an element F ...
Page 1733
... lemma follows , as has been shown above . Q.E.D. Lemma 18 enables us to use the method of proof of Theorem 2 in the neighborhood of the boundary of a domain with smooth boundary . This is carried out in the next two lemmas . 19 LEMMA ...
... lemma follows , as has been shown above . Q.E.D. Lemma 18 enables us to use the method of proof of Theorem 2 in the neighborhood of the boundary of a domain with smooth boundary . This is carried out in the next two lemmas . 19 LEMMA ...
Contents
BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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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 Co(I coefficients complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element essential spectrum 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 kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero