Linear Operators: Spectral theory |
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Page 911
... spaces H. ( cf. Lemma IV.4.19 ) . Theorem 1 may now be applied to the restriction Ta of T to the space . to yield a regular positive measure a vanishing on the complement of o ( T ) ( and thus on the complement of o ( T ) ) and a ...
... spaces H. ( cf. Lemma IV.4.19 ) . Theorem 1 may now be applied to the restriction Ta of T to the space . to yield a regular positive measure a vanishing on the complement of o ( T ) ( and thus on the complement of o ( T ) ) and a ...
Page 1173
... space and X * its adjoint space . Let 1 < p , q < ∞ , p1 + q1 = 1. Let ( S , E , μ ) be a positive measure space . Then ( 42 ) sup 9 € L , ( X * ) , \\ , ≤1 | f g ( 6 ) f ( 6 ) } u ( d $ = If fЄ L2 ( X ) . PROOF . That the right side ...
... space and X * its adjoint space . Let 1 < p , q < ∞ , p1 + q1 = 1. Let ( S , E , μ ) be a positive measure space . Then ( 42 ) sup 9 € L , ( X * ) , \\ , ≤1 | f g ( 6 ) f ( 6 ) } u ( d $ = If fЄ L2 ( X ) . PROOF . That the right side ...
Page 1210
... space L ( S , E , v ) where ( S , E , v ) is a positive measure space . Let every element in ( T " ) be v - essentially bounded on each set in an increasing sequence of sets of finite measure which covers S. Then Hypothesis 7 is ...
... space L ( S , E , v ) where ( S , E , v ) is a positive measure space . Let every element in ( T " ) be v - essentially bounded on each set in an increasing sequence of sets of finite measure which covers S. Then Hypothesis 7 is ...
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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