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. 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. 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 ( σ ( 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 ( σ ( 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 ( { } ) of { u } -null elements is a linear subspace of L2 ( { μ ,; } ) . We shall follow the ...
... elements is a { μ ,, } - null element . Since a scalar multiple of a { u } -null element is evidently a { u } -null element , the family N ( { } ) of { u } -null elements is a linear subspace of L2 ( { μ ,; } ) . We shall follow the ...
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