Linear Operators: Spectral theory |
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Page 1223
... domain ? A natural first guess is to choose as domain the collection D , 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 D , 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 1249
... domain of P. Then the identity | a + v | 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 whose ...
... domain of P. Then the identity | a + v | 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 whose ...
Page 1669
... domain in E " , and let I2 be a domain in E " . Let M : I1 → I be a mapping of I , 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 ) , ( i ) for each ...
... domain in E " , and let I2 be a domain in E " . Let M : I1 → I be a mapping of I , 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 ) , ( i ) for each ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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A₁ 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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero