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 2 = f ' " 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 ) = f if ' ( t ) g ( t ) dt 2 = f ' " f ( t ) ig ' ( t ) ...
Page 1249
... domain of P. Then the identity lx + v2 = | Px + Pv2 shows that ( x , v ) + ( v , x ) = ( Px , Pv ) + ( Pv , Pa ) ... 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 lx + v2 = | Px + Pv2 shows that ( x , v ) + ( v , x ) = ( Px , Pv ) + ( Pv , Pa ) ... 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 ...
Page 1669
... domain in E " , and let I be a domain in E " . Let M : I1 → I2 be a mapping of I1 into I such that ( a ) M - 1C is a compact subset of I , whenever C is a compact subset of I2 ; Then ( b ) ( M ( ) ) , e C ( I1 ) , j = 1 , ... , No. 2 ...
... domain in E " , and let I be a domain in E " . Let M : I1 → I2 be a mapping of I1 into I such that ( a ) M - 1C is a compact subset of I , whenever C is a compact subset of I2 ; Then ( b ) ( M ( ) ) , e C ( I1 ) , j = 1 , ... , No. 2 ...
Contents
BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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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 complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions 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 kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero