Linear Operators: Spectral theory |
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Page 984
... dense in this space , and from the Plancherel theorem that the set of all ƒ in L2 ( R ) for which ƒ vanishes except on a compact set in R is dense in L2 ( R ) . Since the map [ f , g ] → fg takes L2 ( R ) × L2 ( R ) onto all of L1 ( R ) ...
... dense in this space , and from the Plancherel theorem that the set of all ƒ in L2 ( R ) for which ƒ vanishes except on a compact set in R is dense in L2 ( R ) . Since the map [ f , g ] → fg takes L2 ( R ) × L2 ( R ) onto all of L1 ( R ) ...
Page 1188
... dense in then the domain D ( T * ) consists , by definition , of all y in for which ( Tx , y ) is continuous for x in D ( T ) . Since D ( T ) is dense in there is ( IV.4.5 ) a uniquely determined point y * in such that ( Tx , y ) = ( x ...
... dense in then the domain D ( T * ) consists , by definition , of all y in for which ( Tx , y ) is continuous for x in D ( T ) . Since D ( T ) is dense in there is ( IV.4.5 ) a uniquely determined point y * in such that ( Tx , y ) = ( x ...
Page 1271
... dense in . Then if x is in D ( T ) , we have | ( T + iI ) x | 2 = ( Tx , Tx ) i ( x , Tx ) ± i ( Tx , x ) + ( x , x ) This shows that if ( Til ) x = = Tx2 + x2 ≥ x2 . 0 , then a = O and so the operators Til have inverses . Let V be the ...
... dense in . Then if x is in D ( T ) , we have | ( T + iI ) x | 2 = ( Tx , Tx ) i ( x , Tx ) ± i ( Tx , x ) + ( x , x ) This shows that if ( Til ) x = = Tx2 + x2 ≥ x2 . 0 , then a = O and so the operators Til have inverses . Let V be the ...
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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