Linear Operators: Spectral theory |
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Page 1000
... clear . Unfortunately it is not clear that the sequence f , is uniformly con- vergent on any region containing an interval of the real axis and so an additional argument is needed . Let U be the open interval ( a , b ) and Q the ...
... clear . Unfortunately it is not clear that the sequence f , is uniformly con- vergent on any region containing an interval of the real axis and so an additional argument is needed . Let U be the open interval ( a , b ) and Q the ...
Page 1652
... clear that if | J | ≤ k , the sequence { F } converges to some F , in L2 ( I ) . Let p be in Co ( I ) . Then - [ F ... clear that | Fn -F ( k ) → 0 as no . This proves that H * ) ( I ) is complete . That it is a Hilbert space follows ...
... clear that if | J | ≤ k , the sequence { F } converges to some F , in L2 ( I ) . Let p be in Co ( I ) . Then - [ F ... clear that | Fn -F ( k ) → 0 as no . This proves that H * ) ( I ) is complete . That it is a Hilbert space follows ...
Page 1689
... clear from ( i ) that { m } is a Cauchy sequence in L ( I ) for J≤k , so that there exist functions g , g in L , ( I ) such that limm → ∞ ' fm - gp = 0 and limfm - gp = 0. It is then clear from Definition 3.26 that limfmg and limfm ...
... clear from ( i ) that { m } is a Cauchy sequence in L ( I ) for J≤k , so that there exist functions g , g in L , ( I ) such that limm → ∞ ' fm - gp = 0 and limfm - gp = 0. It is then clear from Definition 3.26 that limfmg and limfm ...
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