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 SS , TnT in the strong operator topology . Then Sn + Tn → 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 S * operator ...
... operators in Hilbert space with SS , TnT in the strong operator topology . Then Sn + Tn → 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 S * operator ...
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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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