Linear Operators: Spectral theory |
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Page 922
... topology . Then Sn + T2 → S + T , aSaS , and ST , ST in the strong operator topology . If each S , is normal and S is normal then S * → S * in the strong operator topology . PROOF . The first two statements are obvious . The uniform ...
... topology . Then Sn + T2 → S + T , aSaS , and ST , ST in the strong operator topology . If each S , is normal and S is normal then S * → S * in the strong operator topology . PROOF . The first two statements are obvious . The uniform ...
Page 1420
... topology of the Hilbert space D ( T1 ( 7 ) ) is the same as its relative topology as a subspace of the Hilbert space D ( T1 ( 7 + 7 ' ) ) . Indeed , let { f } be a sequence in D ( T1 ( 7 ) ) . Suppose that { f } converges to zero in the ...
... topology of the Hilbert space D ( T1 ( 7 ) ) is the same as its relative topology as a subspace of the Hilbert space D ( T1 ( 7 + 7 ' ) ) . Indeed , let { f } be a sequence in D ( T1 ( 7 ) ) . Suppose that { f } converges to zero in the ...
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
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 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