Linear Operators: Spectral theory |
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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 ) = f ' if ( t ) g ( t ) dt = f'f ( t ) ig ′ ( t ) dt ...
... 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 ) = f ' if ( t ) g ( t ) dt = f'f ( t ) ig ′ ( t ) dt ...
Page 1248
... domain of P and PM ( = PH ) is called the final domain of P. = 5 LEMMA . A bounded linear operator P in Hilbert space is a partial isometry if and only if P * P is a projection . In this case PP * is also a projection and the ranges of ...
... domain of P and PM ( = PH ) is called the final domain of P. = 5 LEMMA . A bounded linear operator P in Hilbert space is a partial isometry if and only if P * P is a projection . In this case PP * is also a projection and the ranges of ...
Page 1249
... domain . of P. Then the identity | x + v2 = | Px + Pv2 shows that ( x , v ) + ( v , x ) ( Px , Pv ) + ( Pv , Px ) ... 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 whose ...
... domain . of P. Then the identity | x + v2 = | Px + Pv2 shows that ( x , v ) + ( v , x ) ( Px , Pv ) + ( Pv , Px ) ... 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 whose ...
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 Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R 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 satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology tr(T unique unitary vanishes vector zero