## Linear Operators: Spectral theory |

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Page 1224

( b ) If T , is

, every self adjoint extension of Tı , satisfies T , CT , CT CT * Proof . If T , CT , and

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 , and

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

...

Page 1236

Every closed

( T * ) determined by a

, ... , k . Conversely , every such restriction Ty of T * is a closed

Every closed

**symmetric**extension of T is the restriction of T * to the subspace of D( T * ) determined by a

**symmetric**family of boundary conditions , B / ( x ) = 0 , i = 1, ... , k . Conversely , every such restriction Ty of T * is a closed

**symmetric**...Page 1272

Maximal

then it has proper

are different from zero . A maximal

proper ...

Maximal

**symmetric**operators . If T is a**symmetric**operator with dense domain ,then it has proper

**symmetric**extensions provided both of its deficiency indicesare different from zero . A maximal

**symmetric**operator is one which has noproper ...

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### Other editions - View all

Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert ... Nelson Dunford,Jacob T. Schwartz No preview available - 1988 |

Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert ... Nelson Dunford,Jacob T. Schwartz No preview available - 1988 |

### Common terms and phrases

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