## Linear Operators: Spectral theory |

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Page 922

T2 , are defined . 2 . 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

Then Sn ...

**topology**, i.e. , TnX + Tx for every x in the space upon which the operators T , T1 ,T2 , are defined . 2 . 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**.Then Sn ...

Page 1420

( a ' ) The

a sequence in D ( T1 ( t ) ) . Suppose that { { n } converges to zero in the

of D ...

( a ' ) The

**topology**of the Hilbert space D ( T_ ( ) ) is the same as its relative**topology**as a subspace of the Hilbert space D ( T2 ( t + r ' ) ) . Indeed , let { In } bea sequence in D ( T1 ( t ) ) . Suppose that { { n } converges to zero in the

**topology**of D ...

Page 1921

F ( 1550 ) Subadditive function , definition , ( 618 ) Subbase for a

10 ) criterion for , 1.4.8 ( 11 ) Subspace , of a linear space , ( 36 ) . ( See also

Manifold ) Summability , of Fourier series , IV.14.34–51 ( 361-364 ) general

principle ...

F ( 1550 ) Subadditive function , definition , ( 618 ) Subbase for a

**topology**, I.4.6 (10 ) criterion for , 1.4.8 ( 11 ) Subspace , of a linear space , ( 36 ) . ( See also

Manifold ) Summability , of Fourier series , IV.14.34–51 ( 361-364 ) general

principle ...

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### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Copyright | |

57 other sections not shown

### Common terms and phrases

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 Nauk 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