Linear Operators: Spectral theory |
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Page 889
... Borel sets ( v ) ∞ ∞ Σ E ( 8 ̧ ) x = E ( \ § ; ) x , i = 1 Ε i = 1 xε H. A spectral measure E defined on the Borel sets in the plane and satis- fying ( iv ) for every Borel set 8 and ( v ) for every sequence { 8 } of dis- joint Borel sets ...
... Borel sets ( v ) ∞ ∞ Σ E ( 8 ̧ ) x = E ( \ § ; ) x , i = 1 Ε i = 1 xε H. A spectral measure E defined on the Borel sets in the plane and satis- fying ( iv ) for every Borel set 8 and ( v ) for every sequence { 8 } of dis- joint Borel sets ...
Page 909
... Borel sets B of the complex plane and vanishing on the complement of a bounded set S. One of the simplest examples of a bounded normal operator is the operator T defined by the formula ( Tx ) ( 2 ) = λx ( 2 ) , x ɛ L2 ( S , B , μ ) . It ...
... Borel sets B of the complex plane and vanishing on the complement of a bounded set S. One of the simplest examples of a bounded normal operator is the operator T defined by the formula ( Tx ) ( 2 ) = λx ( 2 ) , x ɛ L2 ( S , B , μ ) . It ...
Page 913
... Borel set e . Using the Lebesgue decomposition theorem ( III.4.14 ) , let { e } be a sequence of Borel sets such that v , ( en ) = 0 , and such that if e is a Borel subset of the complement en of e , and Σ = v , ( e ) = 0 , then ( e ) ...
... Borel set e . Using the Lebesgue decomposition theorem ( III.4.14 ) , let { e } be a sequence of Borel sets such that v , ( en ) = 0 , and such that if e is a Borel subset of the complement en of e , and Σ = v , ( e ) = 0 , then ( e ) ...
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