Linear Operators: Spectral theory |
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Page 889
... operator is always closed ( IX.1.5 ) , every set in the domain of a spectral measure satisfying ( iii ) is ... normal operator T in Hilbert space & determines a spectral measure which is defined on the Boolean algebra B of all ...
... operator is always closed ( IX.1.5 ) , every set in the domain of a spectral measure satisfying ( iii ) is ... normal operator T in Hilbert space & determines a spectral measure which is defined on the Boolean algebra B of all ...
Page 922
... operators in Hilbert space with Sn → S , Tn → T in the strong operator topology . Then Sn + Tn → S + T , aSn → aS , and SpTn → ST in the strong operator topology . If each S , is normal and S is normal then S * → S * in the strong ...
... operators in Hilbert space with Sn → S , Tn → T in the strong operator topology . Then Sn + Tn → S + T , aSn → aS , and SpTn → ST in the strong operator topology . If each S , is normal and S is normal then S * → S * in the strong ...
Page 934
... operators in a Hilbert space , then AB is self adjoint . It has been seen ( cf. Exercise X.8.7 ) that if A and B are commuting positive operators , then AB is positive . If A is a normal operator and if B is an operator which commutes ...
... operators in a Hilbert space , then AB is self adjoint . It has been seen ( cf. Exercise X.8.7 ) that if A and B are commuting positive operators , then AB is positive . If A is a normal operator and if B is an operator which commutes ...
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