## Linear Operators: Spectral theory |

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Page 1224

( b ) If T , is symmetric then every symmetric

, every self adjoint

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

Page 1239

_ _ _ Conversely , let T , be a self adjoint

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

family of linearly independent boundary conditions Bi ( 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 of linearly independent boundary conditions Bi ( 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

...

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### Contents

IX | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Compact Groups | 945 |

Copyright | |

46 other sections not shown

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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