## Linear Operators: Spectral theory |

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Page 922

T1 , T2 , . . . , are defined . 1 LEMMA . Let S , T , Sn , Tn , n 2 1 be bounded linear

operators in Hilbert space with Sn →S , T ' n + T in the strong operator

**topology**, i . e . , Tnx → Tx for every x in the space upon which the operators T ,T1 , T2 , . . . , are defined . 1 LEMMA . Let S , T , Sn , Tn , n 2 1 be bounded linear

operators in Hilbert space with Sn →S , T ' n + T in the strong operator

**topology**.Page 1420

( a ' ) The

sequence in D ( T1 ( T ) ) . Suppose that { fr } converges to zero in the

D ...

( a ' ) The

**topology**of the Hilbert space D ( T1 ( T ) ) is the same as its relative**topology**as a subspace of the Hilbert space D ( T ( T + ' ) ) . Indeed , let { In } be asequence in D ( T1 ( T ) ) . Suppose that { fr } converges to zero in the

**topology**ofD ...

Page 1921

3 - 4 ( 15 - 17 ) Titchmarsh - Kodaira theorem , XIII . 5 . 18 ( 1364 ) Tonelli

theorem , III . 11 . 14 ( 194 )

space , definition , ( 398 ) theorems on representation of Boolean rings and

algebras , 1 . 12 .

3 - 4 ( 15 - 17 ) Titchmarsh - Kodaira theorem , XIII . 5 . 18 ( 1364 ) Tonelli

theorem , III . 11 . 14 ( 194 )

**Topology**, base and subbase for , 1 . 4 . 6 ( 10 )space , definition , ( 398 ) theorems on representation of Boolean rings and

algebras , 1 . 12 .

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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 |

### Common terms and phrases

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