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 p¿ 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 p¿ 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 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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A₁ 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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero