Linear Operators: Spectral theory |
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Page 877
... element y in Y has an inverse in X if and only if it has an inverse in Y. Consequently the spectrum of y as an element of Y is the same as its spectrum as an element of X. PROOF . If y1 exists as an element of Y then , since X and Y ...
... element y in Y has an inverse in X if and only if it has an inverse in Y. Consequently the spectrum of y as an element of Y is the same as its spectrum as an element of X. PROOF . If y1 exists as an element of Y then , since X and Y ...
Page 878
... elements determines the * -iso- morphism uniquely and we are thus led to the following definition . 12 DEFINITION . Let x be an element of a commutative B * -alge- bra and let fe C ( o ( x ) ) . By f ( x ) will be meant the element in B ...
... elements determines the * -iso- morphism uniquely and we are thus led to the following definition . 12 DEFINITION . Let x be an element of a commutative B * -alge- bra and let fe C ( o ( x ) ) . By f ( x ) will be meant the element in B ...
Page 1339
... elements is a { μ ,, } - null element . Since a scalar multiple of a { u ,, } - null element is evidently a { u } -null element , the family N ( { u } ) of { u } -null elements is a linear subspace of L2 ( { μ ,; } ) . We shall follow ...
... elements is a { μ ,, } - null element . Since a scalar multiple of a { u ,, } - null element is evidently a { u } -null element , the family N ( { u } ) of { u } -null elements is a linear subspace of L2 ( { μ ,; } ) . We shall follow ...
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 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 open set 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 tr(T unique unitary vanishes vector zero