## Linear Operators: Spectral theory |

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Page 1246

Nelson Dunford, Jacob T. Schwartz, William G. Bade, Robert G. Bartle. We may

also regard A as a mapping from the

Hi . In this case A is still continuous , for Axli = ( Ax , Ax ) , = ( A2x , x ) , = ( Ax , x ) ...

Nelson Dunford, Jacob T. Schwartz, William G. Bade, Robert G. Bartle. We may

also regard A as a mapping from the

**dense**subspace D ( T ) of H into the spaceHi . In this case A is still continuous , for Axli = ( Ax , Ax ) , = ( A2x , x ) , = ( Ax , x ) ...

Page 1271

Let T be a symmetric operator with domain D ( T )

) , we have | ( T + il ) w | 2 = ( Tx , Tx ) F i ( x , Tx ) + i ( Tx , x ) + ( x , x ) = \ Tx / 2 + \

x2 2 \ x12 . This shows that if ( T£il ) x = 0 , then x = 0 and so the operators T il ...

Let T be a symmetric operator with domain D ( T )

**dense**in H . Then if x is in D ( T) , we have | ( T + il ) w | 2 = ( Tx , Tx ) F i ( x , Tx ) + i ( Tx , x ) + ( x , x ) = \ Tx / 2 + \

x2 2 \ x12 . This shows that if ( T£il ) x = 0 , then x = 0 and so the operators T il ...

Page 1905

9 ( 1226 ) De Morgan , rules of , ( 2 )

298 )

. 4 .

9 ( 1226 ) De Morgan , rules of , ( 2 )

**Dense**convex sets , V . 7 . 27 ( 437 ) ... 19 (298 )

**density**of simple functions in L , 1sp < 00 , III . 3 . 8 ( 125 ) ... 11 ( 21 )**Density**of the natural embedding of a B - space X into X * * in the X * topology , V. 4 .

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 869 |

Commutative BAlgebras | 877 |

Copyright | |

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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 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 unit vanishes vector zero