## Linear Operators: Spectral theory |

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Page 1218

If the

... for there is an open set o containing e and a closed set 8 contained in e with

ulo - 8 ) < E . Clearly the

If the

**restrictions**flo , g | d are continuous then so is the**restriction**( af + Bg ) lo n 8... for there is an open set o containing e and a closed set 8 contained in e with

ulo - 8 ) < E . Clearly the

**restriction**of me to the complement of o - is continuous .Page 1239

Conversely , let T , be a self adjoint extension of T . Then by Lemma 26 , T , is the

linearly independent boundary conditions B ; ( x ) = 0 , i = 1 , . . . , k , and we ...

Conversely , let T , be a self adjoint extension of T . Then by Lemma 26 , T , is the

**restriction**of T * to a subspace W of D ( T * ) determined by a symmetric family oflinearly independent boundary conditions B ; ( x ) = 0 , i = 1 , . . . , k , and we ...

Page 1471

31 , a set of boundary conditions defining a self adjoint

the form B ( f ) = QG ( 1 ) + zG2 ( t ) = 0 , ai + až 0 , Qq , ką real , B ( 1 ) = B , G ( f )

+ B2G2 ( ) = 0 , $ i + B2 = 0 , B1 , B2 real , if + has boundary values both at a and

...

31 , a set of boundary conditions defining a self adjoint

**restriction**T of Ti ( t ) is ofthe form B ( f ) = QG ( 1 ) + zG2 ( t ) = 0 , ai + až 0 , Qq , ką real , B ( 1 ) = B , G ( f )

+ B2G2 ( ) = 0 , $ i + B2 = 0 , B1 , B2 real , if + has boundary values both at a and

...

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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