Linear Operators, Part 2 |
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Page 1223
... domain ? A natural first guess is to choose as domain the collection D1 of all functions with one con- tinuous derivative . If ƒ and g are any two such functions , we have ( iDf , g ) = [ ' , if ' ( t ) g ( t ) dt = [ ' , f ( t ) ig ...
... domain ? A natural first guess is to choose as domain the collection D1 of all functions with one con- tinuous derivative . If ƒ and g are any two such functions , we have ( iDf , g ) = [ ' , if ' ( t ) g ( t ) dt = [ ' , f ( t ) ig ...
Page 1249
... domain of P. Then the identity a + v2 = Px + Pv2 shows that ( x , v ) + ( v , x ) ( Px , Pv ) + ( Pv , Px ) . Hence ... domain is dense , then T can be written in one and only one way as a product T = PA , where P is a partial isometry ...
... domain of P. Then the identity a + v2 = Px + Pv2 shows that ( x , v ) + ( v , x ) ( Px , Pv ) + ( Pv , Px ) . Hence ... domain is dense , then T can be written in one and only one way as a product T = PA , where P is a partial isometry ...
Page 1669
... domain in E " and let I be a domain in E2 . Let M : 1 , →→ I be a mapping of I , into I , satisfying the hypotheses 11 ( a ) and ( b ) of Definition 44. Then 2 2 ( a ) the mapping F → FM - 1 is a continuous linear mapping of D ( I1 ) ...
... domain in E " and let I be a domain in E2 . Let M : 1 , →→ I be a mapping of I , into I , satisfying the hypotheses 11 ( a ) and ( b ) of Definition 44. Then 2 2 ( a ) the mapping F → FM - 1 is a continuous linear mapping of D ( I1 ) ...
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 domain 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(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma 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