## Linear Operators: Spectral theory |

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Page 1241

Nelson Dunford, Jacob T. Schwartz. then , letting 2n = xm - Yn , we have limn -

702n = 0 and limm , n70012m - nl + = 0 .

that Izml + SM , m = 1 , 2 , . . . . Moreover , given a > 0 there is an integer N such ...

Nelson Dunford, Jacob T. Schwartz. then , letting 2n = xm - Yn , we have limn -

702n = 0 and limm , n70012m - nl + = 0 .

**Consequently**there is a number M suchthat Izml + SM , m = 1 , 2 , . . . . Moreover , given a > 0 there is an integer N such ...

Page 1383

With boundary conditions A , the eigenvalues are

from the equation sin vă = 0 .

the numbers of the form ( na ) ? , n 2 1 ; in Case C , the numbers { ( n + 1 ) a } ? , n

...

With boundary conditions A , the eigenvalues are

**consequently**to be determinedfrom the equation sin vă = 0 .

**Consequently**, in Case A , the eigenvalues 1 arethe numbers of the form ( na ) ? , n 2 1 ; in Case C , the numbers { ( n + 1 ) a } ? , n

...

Page 1473

to Lemma XII . 3 . 6 ( d ) , that ( T - 2 ) D ( T ) is dense in L ( 1 ) for every real 2 .

Now let – 00 < a < ho . By Definition 6 . 1 , ( T - 21 ) D ( T ) = L ( 1 ) . Thus , 21 - T

is a ...

**Consequently**, T * f = af has no solutions for any real 2 . This means , accordingto Lemma XII . 3 . 6 ( d ) , that ( T - 2 ) D ( T ) is dense in L ( 1 ) for every real 2 .

Now let – 00 < a < ho . By Definition 6 . 1 , ( T - 21 ) D ( T ) = L ( 1 ) . Thus , 21 - T

is a ...

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### Contents

IX | 859 |

extensive presentation of applications of the spectral theorem | 911 |

Miscellaneous Applications | 937 |

Copyright | |

20 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 adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f give given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero