Linear Operators: Spectral theory |
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Page 1191
... adjoint of id / dt . The problem , suggested by the preceding example , of finding self adjoint extensions of a given symmetric operator will be treated systematically in Section 4 . 2. The Spectral Theorem for Unbounded Self Adjoint ...
... adjoint of id / dt . The problem , suggested by the preceding example , of finding self adjoint extensions of a given symmetric operator will be treated systematically in Section 4 . 2. The Spectral Theorem for Unbounded Self Adjoint ...
Page 1247
... adjoint transformation has a unique positive " square root ” . 3 LEMMA . If T is a positive self adjoint transformation , there is a unique positive self adjoint transformation A such that A2 = = T. = PROOF . By Lemma 2 , o ( T ) ≤ [ 0 ...
... adjoint transformation has a unique positive " square root ” . 3 LEMMA . If T is a positive self adjoint transformation , there is a unique positive self adjoint transformation A such that A2 = = T. = PROOF . By Lemma 2 , o ( T ) ≤ [ 0 ...
Page 1290
... adjoint provided only that the coefficients pi are real . In the same way , the formal differential operator ( i / 2 ) ( d / dt ) " { p ( t ) ( d / dt ) + ( d / dt ) p ( t ) } ( d / dt ) " is formally self adjoint provided that p ( t ) ...
... adjoint provided only that the coefficients pi are real . In the same way , the formal differential operator ( i / 2 ) ( d / dt ) " { p ( t ) ( d / dt ) + ( d / dt ) p ( t ) } ( d / dt ) " is formally self adjoint provided that p ( t ) ...
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