Linear Operators: Spectral theory |
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Page 1175
5 Then X is a bounded mapping of the space L ( L , ( S ) ) into itself . Proof . For each real 5o , let H to be the mapping in L , ( L , ( S ) ) 50 defined by the formula ( 47 ) ( 6.1 ) ( 5 ) = f ( 5 ) , > $ o = 0 otherwise .
5 Then X is a bounded mapping of the space L ( L , ( S ) ) into itself . Proof . For each real 5o , let H to be the mapping in L , ( L , ( S ) ) 50 defined by the formula ( 47 ) ( 6.1 ) ( 5 ) = f ( 5 ) , > $ o = 0 otherwise .
Page 1669
Let I , be a domain in Eại , and let I , domain in Ens . Let M : 11 +1 , be a mapping of l , into I , such that ( a ) M - C is a compact subset of I , whenever C is a compact subset of 12 ; ( b ) ( M. ) ) ; Co ( 11 ) , j = 1 , ...
Let I , be a domain in Eại , and let I , domain in Ens . Let M : 11 +1 , be a mapping of l , into I , such that ( a ) M - C is a compact subset of I , whenever C is a compact subset of 12 ; ( b ) ( M. ) ) ; Co ( 11 ) , j = 1 , ...
Page 1736
The mapping g = $ gC is a continuous mapping of HP ) ( 8-11 ) into HP ( C ) by Lemmas 3.22 and 3.23 , and evidently maps C ( 1 ) into C ( C ) . It follows from Definition 3.15 that it maps HP8-11 ) into H . ” ( C ) .
The mapping g = $ gC is a continuous mapping of HP ) ( 8-11 ) into HP ( C ) by Lemmas 3.22 and 3.23 , and evidently maps C ( 1 ) into C ( C ) . It follows from Definition 3.15 that it maps HP8-11 ) into H . ” ( C ) .
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Contents
8 | 876 |
859 | 885 |
extensive presentation of applications of the spectral theorem | 911 |
Copyright | |
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additive adjoint adjoint operator algebra analytic assume basis belongs Borel set boundary conditions boundary values bounded called clear closed closure commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues elements equal equation equivalent Exercise exists extension fact finite dimensional follows follows from Lemma formal differential operator formula function given Hence Hilbert space Hilbert-Schmidt ideal identity immediately implies independent inequality integral interval invariant isometric isomorphism Lemma limit linear Ly(R mapping matrix measure multiplicity neighborhood norm obtained orthonormal positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown shows solutions spectral spectrum square-integrable statement subset subspace sufficient Suppose symmetric Theorem theory topology transform uniformly unique unit unitary vanishes vector zero