Linear Operators: Spectral theory |
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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 1245
... dense domain in Hilbert space has a unique factorization T PA , where A is a positive ( i.e. , ( Ax , x ) ≥ 0 , x ɛ ... dense domain . Then tion ; ( a ) D ( T ) is dense and T ** = T ; ( b ) ( I + T * T ) -1 exists and is a bounded self ...
... dense domain in Hilbert space has a unique factorization T PA , where A is a positive ( i.e. , ( Ax , x ) ≥ 0 , x ɛ ... dense domain . Then tion ; ( a ) D ( T ) is dense and T ** = T ; ( b ) ( I + T * T ) -1 exists and is a bounded self ...
Page 1271
... dense in H. Then if x is in D ( T ) , we have | ( Til ) x2 = ( Tx , Tx ) i ( x , Tx ) + i ( Tx , x ) + ( x , x ) = x = Tx2 + x2 ≥ x2 . This shows that if ( Ti ) x = 0 , then a 0 and so the operators Til have inverses . Let V be the ...
... dense in H. Then if x is in D ( T ) , we have | ( Til ) x2 = ( Tx , Tx ) i ( x , Tx ) + i ( Tx , x ) + ( x , x ) = x = Tx2 + x2 ≥ x2 . This shows that if ( Ti ) x = 0 , then a 0 and so the operators Til have inverses . Let V be the ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T transform unique unitary vanishes vector zero