## Linear Operators: Spectral theory |

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Page 1278

CHAPTER XIII Ordinary Differential Operators 1 . Introduction : Elementary

Properties of

the most important single class of operators are the differential operators . The

study ...

CHAPTER XIII Ordinary Differential Operators 1 . Introduction : Elementary

Properties of

**Formal Differential Operators**From the point of view of applications ,the most important single class of operators are the differential operators . The

study ...

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A

1 7 = 2 , ( 0 ) ( ) * + 0 , - 16 ) ( ) + . . . + aolt ) , = 3a , 40 ) ( ) i = = 0 such that the

complex - valued functions ai , called the coefficient functions , belong to Co ( I ) ...

A

**formal differential operator**of order n on the interval I is an expression din Id n -1 7 = 2 , ( 0 ) ( ) * + 0 , - 16 ) ( ) + . . . + aolt ) , = 3a , 40 ) ( ) i = = 0 such that the

complex - valued functions ai , called the coefficient functions , belong to Co ( I ) ...

Page 1290

In the same way , the

( d / dt ) p ( t ) } ( d / dt ) " is formally self adjoint provided that p ( t ) is a real

function . If we use these observations inductively , we can give a closed form for

the ...

In the same way , the

**formal differential operator**( i / 2 ) ( d / dt ) " { p ( t ) ( d / dt ) +( d / dt ) p ( t ) } ( d / dt ) " is formally self adjoint provided that p ( t ) is a real

function . If we use these observations inductively , we can give a closed form for

the ...

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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