Linear Operators: Spectral theory |
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Page 1215
... preceding proof we let F that , by Lemma 9 , [ TM , ( Uƒ ) a ( 2 ) Wa ( ' , 2 ) μa ( d2 ) = E ( [ - n , n ] ) F ( T ) a → F ( T ) a = U¿1F = fa • ∞ a = ( Uf ) a so Thus the integral ƒ ‰ ( Uƒ ) a ( 2 ) W。( s , λ ) μ ( dλ ) exists in ...
... preceding proof we let F that , by Lemma 9 , [ TM , ( Uƒ ) a ( 2 ) Wa ( ' , 2 ) μa ( d2 ) = E ( [ - n , n ] ) F ( T ) a → F ( T ) a = U¿1F = fa • ∞ a = ( Uf ) a so Thus the integral ƒ ‰ ( Uƒ ) a ( 2 ) W。( s , λ ) μ ( dλ ) exists in ...
Page 1437
... preceding lemma that 20 € σ ( T1 ( 7 ) ) , so that by Definition 6.1 , 2 € 0 , ( T ) . = Conversely , let o eo , ( T ) . Let D , be the closure in the Hilbert space ( T ( T ) ) of D ( To ( 7 ) ) , and let To ( 7 ) be the restriction of ...
... preceding lemma that 20 € σ ( T1 ( 7 ) ) , so that by Definition 6.1 , 2 € 0 , ( T ) . = Conversely , let o eo , ( T ) . Let D , be the closure in the Hilbert space ( T ( T ) ) of D ( To ( 7 ) ) , and let To ( 7 ) be the restriction of ...
Page 1474
... preceding lemma that if λ1 , λ2 € J , and λ1 < λ < Â1⁄2 , then λ € J. Thus J , is an interval . Our second assertion follows immediately from the preceding lemma . Q.E.D. 45 LEMMA . At most one eigenvalue of T lies in any interval J ...
... preceding lemma that if λ1 , λ2 € J , and λ1 < λ < Â1⁄2 , then λ € J. Thus J , is an interval . Our second assertion follows immediately from the preceding lemma . Q.E.D. 45 LEMMA . At most one eigenvalue of T lies in any interval J ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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