## Linear Operators, Part 2 |

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Page 885

If { Bm } is a generalized sequence in Ao converging in the weak operator

the centralizer of its centralizer if and only if it is closed in the weak operator

If { Bm } is a generalized sequence in Ao converging in the weak operator

**topology**to an operator B , then for every A A and x ... in Hilbert space is equal tothe centralizer of its centralizer if and only if it is closed in the weak operator

**topology**.Page 922

T2 , ... , are defined . 1 LEMMA . Let S , T , S. , T. , n 21 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 , S. , T. , n 21 be bounded linear

operators in Hilbert space with Sn → S , Tn → T in the strong operator

**topology**.Page 1921

... Liouville operator , XIII.2 ( 1291 ) , XIII.9.F ( 1550 ) Subadditive function ,

definition , ( 618 ) Subbase for a

Subspace , of a linear space , ( 36 ) . ( See also Manifold ) Summability , of

Fourier series ...

... Liouville operator , XIII.2 ( 1291 ) , XIII.9.F ( 1550 ) Subadditive function ,

definition , ( 618 ) Subbase for a

**topology**, 1.4.6 ( 10 ) criterion for , 1.4.8 ( 11 )Subspace , of a linear space , ( 36 ) . ( See also Manifold ) Summability , of

Fourier series ...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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