Linear Operators: Spectral theory |
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Page 1174
... f ( s ) ds = [ h ( s ) | f ( s ) \ ds ; hence ( 42 ) follows from the similar well - known equation for scalar- valued functions . This concludes the proof of Lemma 21 , and with it the proof of Theorem 20. Q.E.D. 22 COROLLARY . Let ( S ...
... f ( s ) ds = [ h ( s ) | f ( s ) \ ds ; hence ( 42 ) follows from the similar well - known equation for scalar- valued functions . This concludes the proof of Lemma 21 , and with it the proof of Theorem 20. Q.E.D. 22 COROLLARY . Let ( S ...
Page 1656
... Let I be an open set in E " . Let I。 be an open subset of I and let k be an integer . Then ( i ) the correspondence FF is a continuous mapping of A ( * ) ( I ) into itself and of H ( I ) into itself ; ( ii ) the correspondence F → FI ...
... Let I be an open set in E " . Let I。 be an open subset of I and let k be an integer . Then ( i ) the correspondence FF is a continuous mapping of A ( * ) ( I ) into itself and of H ( I ) into itself ; ( ii ) the correspondence F → FI ...
Page 1696
... F to a distribution in D ( D ) such that the carrier of G is C , and a sequence of elements Îm Îm m in Co ( D ) such that m → G as moo . Let y be in Co ( I ) and have ( x ) = 1 for all x in a neighborhood of C. Then mym is in Co ( I ) ...
... F to a distribution in D ( D ) such that the carrier of G is C , and a sequence of elements Îm Îm m in Co ( D ) such that m → G as moo . Let y be in Co ( I ) and have ( x ) = 1 for all x in a neighborhood of C. Then mym is in Co ( I ) ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Copyright | |
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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