Linear Operators, Part 2 |
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Page 1649
... let F be a distribution in I. Then the distribution F in I defined by the equation F ( q ) = F ( 9 ) , 9 € Co ( I ) , is called the complex conjugate of F. 8 LEMMA . Let I and F be as in the preceding definition , and let t be a formal ...
... let F be a distribution in I. Then the distribution F in I defined by the equation F ( q ) = F ( 9 ) , 9 € Co ( I ) , is called the complex conjugate of F. 8 LEMMA . Let I and F be as in the preceding definition , and let t be a formal ...
Page 1678
... ( F ) K is compact , and a and â are complex numbers , then 1 α F ( xp + âî ) = aF ( y ) + âF ( 4 ) . PROOF . Let be a second function in Co ( I ) such that y ( x ) = 1 for a in a neighborhood of K1 . Then pp - p vanishes in a neigh ...
... ( F ) K is compact , and a and â are complex numbers , then 1 α F ( xp + âî ) = aF ( y ) + âF ( 4 ) . PROOF . Let be a second function in Co ( I ) such that y ( x ) = 1 for a in a neighborhood of K1 . Then pp - p vanishes in a neigh ...
Page 1696
... F to a distribution in D ( D ) such that the carrier of G is C , and a sequence of elements m in Co ( D ) such that m → G as moo . Let y be in Co ( I ) and have p ( x ) - ∞ . 1 for all in a neighborhood of C. Then m = 4Ŷm is in a Co ...
... F to a distribution in D ( D ) such that the carrier of G is C , and a sequence of elements m in Co ( D ) such that m → G as moo . Let y be in Co ( I ) and have p ( x ) - ∞ . 1 for all in a neighborhood of C. Then m = 4Ŷm is in a Co ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm 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 transform unique unitary vanishes vector zero