## Linear Operators: Spectral theory |

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Page 1190

If T is an

Lemma 6 ( a ) T is closed and by the closed graph theorem ( II . 2 . 4 ) , T is

bounded . Thus an

If T is an

**everywhere**defined symmetric operator then T * 2 T and thus T * = T . ByLemma 6 ( a ) T is closed and by the closed graph theorem ( II . 2 . 4 ) , T is

bounded . Thus an

**everywhere**defined symmetric operator is bounded and self ...Page 1212

Then S . ( Bu + 18 ) ( ) F ( 2 ) u ( da ) = ls f ( 8 ) ( An + 1F ) ( s ) v ( ds ) = 15 . 1 ( s ) (

F ( T ) g ) ( s ) v ( ds ) An F ) ( s ) v ( ds ) = S , ( B , 1 ) ( 2 ) F ( ) u ( da ) . Thus , ( Bm

+ 11 ) ( a ) = ( 0 , 1 ) ( 2 ) u - almost

Then S . ( Bu + 18 ) ( ) F ( 2 ) u ( da ) = ls f ( 8 ) ( An + 1F ) ( s ) v ( ds ) = 15 . 1 ( s ) (

F ( T ) g ) ( s ) v ( ds ) An F ) ( s ) v ( ds ) = S , ( B , 1 ) ( 2 ) F ( ) u ( da ) . Thus , ( Bm

+ 11 ) ( a ) = ( 0 , 1 ) ( 2 ) u - almost

**everywhere**on en . Consequently ( B , A ) ...Page 1233

Then ( T2 - 201 ) - 1 = R ( 20 ) is an

norm not more than TI ( 20 ) - 1 . Consequently , the series ( – ho ) " R ( 10 * + 1 n

+ 1 n = 0 converges if 12 - 201 < \ I ( 20 ) ] . Since T , is closed , we have ( 7 , - 21 )

...

Then ( T2 - 201 ) - 1 = R ( 20 ) is an

**everywhere**defined , bounded operator ofnorm not more than TI ( 20 ) - 1 . Consequently , the series ( – ho ) " R ( 10 * + 1 n

+ 1 n = 0 converges if 12 - 201 < \ I ( 20 ) ] . Since T , is closed , we have ( 7 , - 21 )

...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 869 |

Commutative BAlgebras | 877 |

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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