Linear Operators: Spectral theory |
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Page 1669
Let M : 1 +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 ( - ) ) , E CR ( 11 ) , j ... , 12 . Then ( i ) for each q in Co ( 12 ) , qoM will denote the ...
Let M : 1 +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 ( - ) ) , E CR ( 11 ) , j ... , 12 . Then ( i ) for each q in Co ( 12 ) , qoM will denote the ...
Page 1671
If F corresponds to the functions , we have k ( Fo M - 1 ) ( 0 ) F ( po M ) = $ t ( x ) q ( M ( x ) ) dx = st ( M - 1 ( x ) ] p ( x ) J ( x ) dx , = M J denoting the absolute value of the Jacobian determinant of the mapping x → M - 1 ...
If F corresponds to the functions , we have k ( Fo M - 1 ) ( 0 ) F ( po M ) = $ t ( x ) q ( M ( x ) ) dx = st ( M - 1 ( x ) ] p ( x ) J ( x ) dx , = M J denoting the absolute value of the Jacobian determinant of the mapping x → M - 1 ...
Page 1736
The mapping g → $ g C is a continuous mapping of H ( P ) ( € -11 ) into H ( o ) ( C ) by Lemmas 3.22 and 3.23 , and evidently maps Co ( 1 ) into C7 ° ( C ) . It follows from Definition 3.15 that it maps H ” ( 8-11 ) into H ” ( C ) .
The mapping g → $ g C is a continuous mapping of H ( P ) ( € -11 ) into H ( o ) ( C ) by Lemmas 3.22 and 3.23 , and evidently maps Co ( 1 ) into C7 ° ( C ) . It follows from Definition 3.15 that it maps H ” ( 8-11 ) into H ” ( C ) .
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Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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additive Akad algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure multiplicity neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero