## Linear operators. 2. Spectral theory : self adjoint operators in Hilbert Space |

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Page 1224

( b ) If T , is symmetric then every symmetric

, every self adjoint

y e D ( 1 * ) , then ( x , T * y ) = ( Tox , y ) ( T , x , y ) for any x eD ( T2 ) . Hence y eD

...

( b ) If T , is symmetric then every symmetric

**extension**T , of T ,, and , in particular, every self adjoint

**extension**of Tı , satisfies T , CT , CT CT * Proof . If T , CT , andy e D ( 1 * ) , then ( x , T * y ) = ( Tox , y ) ( T , x , y ) for any x eD ( T2 ) . Hence y eD

...

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 B1 ( x ) = 0 , i = 1 , ... , k , and we have ...

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 B1 ( x ) = 0 , i = 1 , ... , k , and we have ...

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

...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

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