## 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 , Tn →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 , Tn →T in the strong operator

**topology**.Page 1420

( a ' ) The

be a sequence in D ( T2 ( ) ) . Suppose that { fr } converges to zero in the

of D ...

( a ' ) The

**topology**of the Hilbert space D ( T ( T ) ) is the same as its relative**topology**as a subspace of the Hilbert space D ( Ti ( 7 + 7 ' ) ) . Indeed , let { { n }be a sequence in D ( T2 ( ) ) . Suppose that { fr } converges to zero in the

**topology**of D ...

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

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 singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero