Linear Operators: Spectral theory |
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Page 925
... sequence of normal operators in H , all commuting with each other . Show that there exists a single Hermitian operator T such that each N is a Borel function of T. ( Hint : Use Theorem 2.1 and Exercise 15 ) . k 17 For operators A , B ...
... sequence of normal operators in H , all commuting with each other . Show that there exists a single Hermitian operator T such that each N is a Borel function of T. ( Hint : Use Theorem 2.1 and Exercise 15 ) . k 17 For operators A , B ...
Page 959
... sequence { eeb , m ≥1 } is an increasing sequence of sets whose union is eb ,. Since uo is countably additive on Bo , Mo ( eb2 ) = limm Mo ( eembn ) ≥k , and so for some m , μo ( eem ) ≥ μo ( еembn ) > k − ɛ . This shows that the ...
... sequence { eeb , m ≥1 } is an increasing sequence of sets whose union is eb ,. Since uo is countably additive on Bo , Mo ( eb2 ) = limm Mo ( eembn ) ≥k , and so for some m , μo ( eem ) ≥ μo ( еembn ) > k − ɛ . This shows that the ...
Page 1124
... sequence contains either a monotone- increasing or a monotone - decreasing sequence , it therefore follows that q ( E ) → q ( E ) implies E → E strongly . Hence , if we choose a countable set { E } CF such that { p ( E ) } is dense in ...
... sequence contains either a monotone- increasing or a monotone - decreasing sequence , it therefore follows that q ( E ) → q ( E ) implies E → E strongly . Hence , if we choose a countable set { E } CF such that { p ( E ) } is dense in ...
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