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 H 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 H there is ( IV.4.5 ) a uniquely determined point y * in such that ( Tx , y ) ( x ...
Page 1271
... dense in H. Then if x is in D ( T ) , we have | ( T ± iI ) x | 2 = ( Tx , Tx ) = i ( x , Tx ) ± i ( Tx , x ) + ( x , x ) - = Tx2 + x2x2 . This shows that if ( Ti ) x 0 , then x = O and so the operators Til have inverses . Let V be the ...
... dense in H. Then if x is in D ( T ) , we have | ( T ± iI ) x | 2 = ( Tx , Tx ) = i ( x , Tx ) ± i ( Tx , x ) + ( x , x ) - = Tx2 + x2x2 . This shows that if ( Ti ) x 0 , then x = O and so the operators Til have inverses . Let V be the ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 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 prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology transform unique unitary vanishes vector zero