Linear Operators: Spectral operators |
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Page 1175
... mapping of the space L ( L ( S ) ) into itself . 1 H PROOF . For each real 50 , let be the mapping in L ( L , ( S ) ) defined by the formula ( 47 ) 5ο ( H 5 , † ) ( § ) ( f ) ( E ) = f ( E ) , § > 50 , = 0 otherwise . By Corollary 22 ...
... mapping of the space L ( L ( S ) ) into itself . 1 H PROOF . For each real 50 , let be the mapping in L ( L , ( S ) ) defined by the formula ( 47 ) 5ο ( H 5 , † ) ( § ) ( f ) ( E ) = f ( E ) , § > 50 , = 0 otherwise . By Corollary 22 ...
Page 1671
... mapping → M - 1 ( x ) ; this follows by the standard theorem on change of variables in a multiple integral . But then ( iii ) is evident . Q.E.D. Lemma 47 allows us to describe the behavior of the spaces H3 , A ” , etc. , under the ...
... mapping → M - 1 ( x ) ; this follows by the standard theorem on change of variables in a multiple integral . But then ( iii ) is evident . Q.E.D. Lemma 47 allows us to describe the behavior of the spaces H3 , A ” , etc. , under the ...
Page 1707
... mapping of Hm + P ) ( C ) into Hm ) ( C ) , and of H + P - 1 ) ( C ) into H - 1 ) ( C ) , is less than min ( m , m - 1 ) . Then , by Lemma VII.3.4 , the mapping Hm ( 1 + 0εti1 ) , π regarded either as a mapping of Hm ) ( C ) or of Him ...
... mapping of Hm + P ) ( C ) into Hm ) ( C ) , and of H + P - 1 ) ( C ) into H - 1 ) ( C ) , is less than min ( m , m - 1 ) . Then , by Lemma VII.3.4 , the mapping Hm ( 1 + 0εti1 ) , π regarded either as a mapping of Hm ) ( C ) or of Him ...
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 linear 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 satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology unique unitary vanishes vector zero