Linear Operators: Spectral theory |
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Page 1303
Clearly B ( l ) = 0 for those which vanish in a neighborhood of a . Thus B is a boundary value for ī at a . To prove the converse , let B be a boundary value at a . Choose a function h in Co ( 1 ) which is identically equal to one in a ...
Clearly B ( l ) = 0 for those which vanish in a neighborhood of a . Thus B is a boundary value for ī at a . To prove the converse , let B be a boundary value at a . Choose a function h in Co ( 1 ) which is identically equal to one in a ...
Page 1403
It is then sufficient to show that each de A has a neighborhood 1 , such that W ; ( :, ) e L ( a , c ) for Mi - almost all à e do , since 1 may then be written as a countable union of such neighborhoods 1 ,. We shall show below that for ...
It is then sufficient to show that each de A has a neighborhood 1 , such that W ; ( :, ) e L ( a , c ) for Mi - almost all à e do , since 1 may then be written as a countable union of such neighborhoods 1 ,. We shall show below that for ...
Page 1733
Q.E.D. Lemma 18 enables us to use the method of proof of Theorem 2 in the neighborhood of the boundary of a domain with smooth boundary . This is carried out in the next two lemmas . 19 LEMMA . Let o be an elliptic formal partial ...
Q.E.D. Lemma 18 enables us to use the method of proof of Theorem 2 in the neighborhood of the boundary of a domain with smooth boundary . This is carried out in the next two lemmas . 19 LEMMA . Let o be an elliptic formal partial ...
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Contents
8 | 876 |
859 | 885 |
extensive presentation of applications of the spectral theorem | 911 |
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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