Linear Operators: Spectral theory |
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Page 958
... disjoint sets in Bo whose union e is also in Bo . Let r1 = en Uen + 1 ... , so that E ( r ) g 0 for every g in L2 ( R ) and , by Lemma 5 , ( g , y ( e , ) ) = 8E ( e ) E ( r , ) g → 0 . This argument shows that the vector valued additive ...
... disjoint sets in Bo whose union e is also in Bo . Let r1 = en Uen + 1 ... , so that E ( r ) g 0 for every g in L2 ( R ) and , by Lemma 5 , ( g , y ( e , ) ) = 8E ( e ) E ( r , ) g → 0 . This argument shows that the vector valued additive ...
Page 959
... sets whose union is eb ,. Since μo is countably additive on Bo , Mo ( ebn ) ... disjoint sequence in B. It is clear that μ ( Uan ) ≥μ ( an ) , so that , if ... disjoint sets in B。 with e = ∞ 9 Uen . Let ɛ > 0 be given , and let N be so ...
... sets whose union is eb ,. Since μo is countably additive on Bo , Mo ( ebn ) ... disjoint sequence in B. It is clear that μ ( Uan ) ≥μ ( an ) , so that , if ... disjoint sets in B。 with e = ∞ 9 Uen . Let ɛ > 0 be given , and let N be so ...
Page 1187
... disjoint and their union is the whole plane . The resolvent set p ( T ) is open and the resolvent R ( 2 ; T ) is an ... sets and that if a point 2 is not in any of these sets the inverse ( λI — T ) -1 must exist as an everywhere defined ...
... disjoint and their union is the whole plane . The resolvent set p ( T ) is open and the resolvent R ( 2 ; T ) is an ... sets and that if a point 2 is not in any of these sets the inverse ( λI — T ) -1 must exist as an everywhere defined ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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Common terms and phrases
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 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 PROOF prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T unique unitary vanishes vector zero