## Linear Operators: Spectral theory |

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Page 1218

... function f on R and every e > 0 there is a Borel set o in R with u ( o ) < ε and

such that the

thus ...

... function f on R and every e > 0 there is a Borel set o in R with u ( o ) < ε and

such that the

**restriction**of f to the complement of o is continuous . Proof . If the**restrictions**flo , gd are continuous then so is the**restriction**( af + Bg ) lo n d andthus ...

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 1613

A

definition to all functions which satisfy a prescribed set of boundary conditions ,

just as in the Hilbert space theory . While the essential spectrum of t in X is

invariant ...

A

**restriction**of the operator T ( t , x ) is obtained by restricting the domain ofdefinition to all functions which satisfy a prescribed set of boundary conditions ,

just as in the Hilbert space theory . While the essential spectrum of t in X is

invariant ...

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

15 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 Akad algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f 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 Russian satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero