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 { eembn , m ≥1 } is an increasing sequence of sets whose union is eb . Since μo is countably additive on Bo , Mo ( eb , ) = limm Mo ( eembn ) k , and so for some m , po ( eem ) ≥μo ( еembn ) > k - e . This shows that the set ...
... sequence { eembn , m ≥1 } is an increasing sequence of sets whose union is eb . Since μo is countably additive on Bo , Mo ( eb , ) = limm Mo ( eembn ) k , and so for some m , po ( eem ) ≥μo ( еembn ) > k - e . This shows that the set ...
Page 1124
... sequence contains either a monotone- increasing or a monotone - decreasing sequence , it therefore follows that ( En ) → q ( E ) implies E → E strongly . Hence , if we choose a countable set { E } CF such that { ( E , ) } is dense in ...
... sequence contains either a monotone- increasing or a monotone - decreasing sequence , it therefore follows that ( En ) → q ( E ) implies E → E strongly . Hence , if we choose a countable set { E } CF such that { ( E , ) } is dense in ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 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 prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology transform unique unitary vanishes vector zero