## Linear Operators: Spectral theory |

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Page 1420

If A is considered as a subset of ý , then the restriction of Ti ( t ' ) to A is a

continuous mapping of A into H . Then , assuming that ttt ' has a non -

leading coefficient , ( A ) the Hilbert spaces D ( T2 ( t + t ' ) ) and D ( T1 ( T ) ) have

the same ...

If A is considered as a subset of ý , then the restriction of Ti ( t ' ) to A is a

continuous mapping of A into H . Then , assuming that ttt ' has a non -

**zero**leading coefficient , ( A ) the Hilbert spaces D ( T2 ( t + t ' ) ) and D ( T1 ( T ) ) have

the same ...

Page 1432

Suppose first that the end point under consideration is finite so that without loss of

generality we can suppose it to be at

by the leading coefficient an of t , we can write the equation ( 7 - 2 ) } = 0 in the ...

Suppose first that the end point under consideration is finite so that without loss of

generality we can suppose it to be at

**zero**. Then , dividing through if necessaryby the leading coefficient an of t , we can write the equation ( 7 - 2 ) } = 0 in the ...

Page 1463

Since all the terms in the integral on the right are non - negative , we must have

tita - fatı identically

Since all the terms in the integral on the right are non - negative , we must have

tita - fatı identically

**zero**in [ c , d ] . Thus ( hta ? ) ' = tz " ( ita - tata ) is identically**zero**in [ c , d ] , so that fit is constant . Moreover , since fı and fí have only a finite ...### What people are saying - Write a review

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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