Linear Operators: Spectral theory |
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Page 925
... sequence of normal operators in § , 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 ) . 17 For operators A , B ...
... sequence of normal operators in § , 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 ) . 17 For operators A , B ...
Page 959
... sequence { eemb , m ≥1 } is an increasing sequence of sets whose union is eb . Since μo is countably additive on Bo , μo ( ebn ) = limm Mo ( eembn ) ≥k , and so for some m , Mo ( eem ) ≥μo ( eembn ) > k - e . This shows that the set ...
... sequence { eemb , m ≥1 } is an increasing sequence of sets whose union is eb . Since μo is countably additive on Bo , μo ( ebn ) = limm Mo ( eembn ) ≥k , and so for some m , Mo ( eem ) ≥μo ( eembn ) > k - e . This shows that the set ...
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 { ( 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 { ( E ) } is dense in ...
Contents
BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element essential spectrum exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero