## Linear Operators: Spectral theory |

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Page 1378

matrix measure { fus } , i , j = 1 , . . . , k of

= 1 , . . . , k ; Pij = 0 , if i > k or j > k . PROOF . Suppose that 01 , . . . , 07 is a

determining set for T . Then it is evident from

i , j ...

matrix measure { fus } , i , j = 1 , . . . , k of

**Theorem**23 is unique , and Pin = Pijo i , j= 1 , . . . , k ; Pij = 0 , if i > k or j > k . PROOF . Suppose that 01 , . . . , 07 is a

determining set for T . Then it is evident from

**Theorem**23 that if we define { Pis } ,i , j ...

Page 1379

Pu } is the matrix measure of

determined for each e Ç N . Since 1 is the union of a sequence of neighborhoods

of the same type as N , the uniqueness of { Ô is } follows immediately . Q . E . D .

27 ...

Pu } is the matrix measure of

**Theorem**23 , the values Pis ( e ) are uniquelydetermined for each e Ç N . Since 1 is the union of a sequence of neighborhoods

of the same type as N , the uniqueness of { Ô is } follows immediately . Q . E . D .

27 ...

Page 1904

15 remarks on , ( 389 - 392 ) Convergence

on convergence of measures , IV . 9 . 15 ( 316 ) Arzelà

limits , IV . 6 . 11 ( 268 ) Banach

15 remarks on , ( 389 - 392 ) Convergence

**theorems**, IV . 15 Alexandroff**theorem**on convergence of measures , IV . 9 . 15 ( 316 ) Arzelà

**theorem**on continuouslimits , IV . 6 . 11 ( 268 ) Banach

**theorem**for operators into space of measurable ...### What people are saying - Write a review

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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