## Linear Operators: Spectral theory |

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Page 915

Hilbert space H and let E

sequence { xn } CH such that H = 2 : 1 H ( x ; ) , where Þ ( x ; ) = sp { f ( T ) xilfe Clo

( T ) ) } , and a decreasing sequence { en } of Borel sets such that ( E ( e ) xn ...

Hilbert space H and let E

**denote**its resolution of the identity . Then there exists asequence { xn } CH such that H = 2 : 1 H ( x ; ) , where Þ ( x ; ) = sp { f ( T ) xilfe Clo

( T ) ) } , and a decreasing sequence { en } of Borel sets such that ( E ( e ) xn ...

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of the closed set C ; we shall

C ) . Since each projection in the spectral resolution of T and hence each

continuous function of T is a strong limit of linear combinations of the projections

Ei , it ...

of the closed set C ; we shall

**denote**this subspace of L [ 0 , 1 ] by the symbol L (C ) . Since each projection in the spectral resolution of T and hence each

continuous function of T is a strong limit of linear combinations of the projections

Ei , it ...

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In the next few paragraphs t

operator of order n , defined on the interval R = { - 00 < t < + 00 } . Let t have the

form m idli 2 a , ( 1 ) ( ) , and suppose that all the coefficients a ; are periodic and

have ...

In the next few paragraphs t

**denotes**a formally self adjoint formal differentialoperator of order n , defined on the interval R = { - 00 < t < + 00 } . Let t have the

form m idli 2 a , ( 1 ) ( ) , and suppose that all the coefficients a ; are periodic and

have ...

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

15 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 |

### Common terms and phrases

additive adjoint operator 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 Russian satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero