## Linear Operators: Spectral theory |

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Page 1239

Conversely , let T , be a self adjoint

restriction of T * to a subspace W of D ( T * ) determined by a symmetric family of

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 therestriction of T * to a subspace W of D ( T * ) determined by a symmetric family of

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

Page 1270

symmetric operator has a self adjoint

determining whether the spectral theorem may be employed . If the answer to this

...

**Extensions**of symmetric operators . The problem of determining whether a givensymmetric operator has a self adjoint

**extension**is of crucial importance indetermining whether the spectral theorem may be employed . If the answer to this

...

Page 1397

The assertion then is that the null space N of T * is at least k - dimensional . The

method of proof is the following : it will be shown that if the theorem is false , then

a proper symmetric

...

The assertion then is that the null space N of T * is at least k - dimensional . The

method of proof is the following : it will be shown that if the theorem is false , then

a proper symmetric

**extension**T , of T can be constructed whose domain properly...

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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

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### 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