## Linear Operators: Spectral theory |

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Page 922

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

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

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

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

**topology**.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 ( T1 ( ) ) is the same as its relative**topology**as a subspace of the Hilbert space D ( T ( 7 + ? ' ) ) . Indeed , let { { n } bea sequence in D ( T1 ( T ) ) . Suppose that { n } converges to zero in the

**topology**of D ...

Page 1921

6 . 16 ( 272 ) complex case , IV . 6 . 17 ( 274 ) remarks on , ( 383 – 385 ) Strictly

convex B - space , definition , VII . 7 ( 458 ) Strictly convex B - space , definition ,

VII . 7 ( 458 ) Strong operator

.

6 . 16 ( 272 ) complex case , IV . 6 . 17 ( 274 ) remarks on , ( 383 – 385 ) Strictly

convex B - space , definition , VII . 7 ( 458 ) Strictly convex B - space , definition ,

VII . 7 ( 458 ) Strong operator

**topology**, definition , VI . 1 . 2 ( 475 ) properties , V1.

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### Contents

IX | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Compact Groups | 945 |

Copyright | |

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additive adjoint operator algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently consider constant contains continuous converges Corollary corresponding defined Definition denote dense derivatives determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function 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 shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero