## Linear Operators: Spectral theory |

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Page 1378

If the determining set 01 , . . . , One of Theorem 23 were known to be part of a

basis 01 , . . . , On with the properties of Theorem 13 , then the uniqueness of { Ô

is } would follow immediately from the

course ...

If the determining set 01 , . . . , One of Theorem 23 were known to be part of a

basis 01 , . . . , On with the properties of Theorem 13 , then the uniqueness of { Ô

is } would follow immediately from the

**preceding**theorem . Moreover , in thecourse ...

Page 1419

On the interval [ Sit1 , Mi + 1 ] , consider the two functions - | ( t ) and fi ( t ) = + | (

284 + 1 - t ) . We have ( - 1 ) " = 9 ( - 1 ) , tl = 91t , where gı ( t ) = 9 ( 28i + 1 - t ) 2 g

( t ) , since q is monotone decreasing . By the

On the interval [ Sit1 , Mi + 1 ] , consider the two functions - | ( t ) and fi ( t ) = + | (

284 + 1 - t ) . We have ( - 1 ) " = 9 ( - 1 ) , tl = 91t , where gı ( t ) = 9 ( 28i + 1 - t ) 2 g

( t ) , since q is monotone decreasing . By the

**preceding**lemma , - f ( t ) < fi ( t ) in ...Page 1771

... such that lim | u ( t , : ) - | ( : ) = 0 . t > 0 , 62 , to PROOF . Statement ( i ) follows

from the

statement ( ii ) of the

hypotheses of ...

... such that lim | u ( t , : ) - | ( : ) = 0 . t > 0 , 62 , to PROOF . Statement ( i ) follows

from the

**preceding**theorem and Theorem 6 . 23 . Statement ( ii ) follows fromstatement ( ii ) of the

**preceding**theorem , since a function satisfying thehypotheses of ...

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### Contents

IX | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Compact Groups | 945 |

Copyright | |

46 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

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