Linear Operators: Spectral theory |
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Page 898
... resolution of the identity for T. In order to relate this notion of the resolution of the identity with that given in Section 1 we state the following corollary . 6 COROLLARY . If E is the resolution of the identity for the normal ...
... resolution of the identity for T. In order to relate this notion of the resolution of the identity with that given in Section 1 we state the following corollary . 6 COROLLARY . If E is the resolution of the identity for the normal ...
Page 920
... resolution of the identity . The following theorem gives a method for calculating the resolution of the identity for a self adjoint operator T in terms of its resolvent R ( x ; T ) ( xI — T ) -1 . It should be recalled ( Theorem 4.2 ) ...
... resolution of the identity . The following theorem gives a method for calculating the resolution of the identity for a self adjoint operator T in terms of its resolvent R ( x ; T ) ( xI — T ) -1 . It should be recalled ( Theorem 4.2 ) ...
Page 1128
... resolution of T is the strong limit of linear combinations of the projections E. This we do as follows . Let A be the commutative B * -algebra of operators generated by the projections E , and let Д be its spectrum . If is any element ...
... resolution of T is the strong limit of linear combinations of the projections E. This we do as follows . Let A be the commutative B * -algebra of operators generated by the projections E , and let Д be its spectrum . If is any element ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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Common terms and phrases
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 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 real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T unique unitary vanishes vector zero