Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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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 ′ ( t ) ...
... 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 ′ ( t ) ...
Page 1248
... domain of P and PM ( = P§ ) 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 ( = P§ ) 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 | xv | 2 = | 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 ...
... domain . of P. Then the identity | xv | 2 = | 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 ...
Contents
IX | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
Copyright | |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients compact subset complex numbers continuous function converges Corollary deficiency indices Definition denote dense domain eigenvalues element essential spectrum 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(R L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood norm open set open subset orthonormal partial differential operator Plancherel's theorem positive PROOF prove real axis real numbers satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose T₁ T₁(t theory To(t topology unique unitary vanishes vector zero