## Linear Operators: Spectral theory |

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Page 990

A bounded measurable function y on R is in the Ly - closed linear subspace of L (

R ) which is

Conversely , if o is in the Ly - closed linear manifold

A bounded measurable function y on R is in the Ly - closed linear subspace of L (

R ) which is

**determined**by the characters in any neighborhood of its spectral set .Conversely , if o is in the Ly - closed linear manifold

**determined**by the ...Page 1321

The matrices l ' = ( y ) and I ' = ( ' s ) in the preceding theorem are uniquely

Proof . We have seen in the derivation of Theorem 8 that the functions & ; ( t ) and

Bi ( t ) ...

The matrices l ' = ( y ) and I ' = ( ' s ) in the preceding theorem are uniquely

**determined**by the jump equations and by the boundary conditions defining T .Proof . We have seen in the derivation of Theorem 8 that the functions & ; ( t ) and

Bi ( t ) ...

Page 1323

To

numbers an ( t ) and Bi ( t ) we have the n jump ... By symmetry ( Vis ) and ( ' s )

are also

conditions Ez ( K ) ...

To

**determine**the u * + 0 * = ( p * + q * ) - ( u * + v * ) = ( n + k * ) - ( u * + 2 * )numbers an ( t ) and Bi ( t ) we have the n jump ... By symmetry ( Vis ) and ( ' s )

are also

**determined**uniquely by the jump conditions and the boundaryconditions Ez ( K ) ...

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### Contents

IX | 859 |

extensive presentation of applications of the spectral theorem | 911 |

Miscellaneous Applications | 937 |

Copyright | |

20 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

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additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f give given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero