## 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 TiÇT , and

ye D ( 1 * ) , then ( x , 7 * y ) = ( Tox , y ) = ( T , x , y ) for any x € D ( Tı ) . Hence ye ...

( 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 TiÇT , and

ye D ( 1 * ) , then ( x , 7 * y ) = ( Tox , y ) = ( T , x , y ) for any x € D ( Tı ) . Hence ye ...

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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### Contents

IX | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Compact Groups | 945 |

Copyright | |

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