Linear Operators: Spectral operators |
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Page 885
... topology . A PROOF . It is clear from the definition of the centralizer that AC ( A ) and from the lemma that A is closed in the weak operator topology if = ( A ) . To prove the theorem it suffices to show that if A is closed in the ...
... topology . A PROOF . It is clear from the definition of the centralizer that AC ( A ) and from the lemma that A is closed in the weak operator topology if = ( A ) . To prove the theorem it suffices to show that if A is closed in the ...
Page 922
... topology . If each S , is normal and S is normal then S → S * in the strong operator topology . n PROOF . The first two statements are obvious . The uniform boundedness theorem ( II.3.21 ) yields a constant K with | S | ≤K , and thus ...
... topology . If each S , is normal and S is normal then S → S * in the strong operator topology . n PROOF . The first two statements are obvious . The uniform boundedness theorem ( II.3.21 ) yields a constant K with | S | ≤K , and thus ...
Page 1921
... topology , definition , VI.1.2 ( 475 ) properties , VI.9.1–5 ( 511 ) , VI.9.11– 12 ( 512-513 ) Strong topology , in a normed space , II.3.1 ( 59 ) , ( 419 ) Structure space of a B - algebra , IX.2.7 ( 869 ) Sturm - Liouville operator ...
... topology , definition , VI.1.2 ( 475 ) properties , VI.9.1–5 ( 511 ) , VI.9.11– 12 ( 512-513 ) Strong topology , in a normed space , II.3.1 ( 59 ) , ( 419 ) Structure space of a B - algebra , IX.2.7 ( 869 ) Sturm - Liouville operator ...
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 Haar measure 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 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 unique unitary vanishes vector zero