## Linear Operators: Spectral theory |

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Page 1027

Hence a

scalar

have ETX = 2x . Then Tx = 2x + y , where y

...

Hence a

**belongs**to the spectrum of ET . Conversely , suppose that a non - zeroscalar

**belongs**to the spectrum of ET . Then , for some non - zero x in EH , wehave ETX = 2x . Then Tx = 2x + y , where y

**belongs**to the subspace ( I - EH , and...

Page 1116

Let the operator B be defined by Bq ; = ( TP : ly ! p / 2 ) - 1 . Then plainly Bq . ? = (

yplej ? < 0o , so that , by Definition 6 . 1 , B

C2 . If we let Aq ; = y ! - p / 20 ; , then A is plainly self adjoint and A

...

Let the operator B be defined by Bq ; = ( TP : ly ! p / 2 ) - 1 . Then plainly Bq . ? = (

yplej ? < 0o , so that , by Definition 6 . 1 , B

**belongs**to the Hilbert - Schmidt classC2 . If we let Aq ; = y ! - p / 20 ; , then A is plainly self adjoint and A

**belongs**to the...

Page 1747

Each such function /

Passing without loss of generality from t to rth , we may assume that 0 € 0 ( V ) .

Let Vi = Vm , where we choose m so large that 2pm 2 [ n / 2 ] + 1 . Then , by ...

Each such function /

**belongs**to the intersection of Co ( 1 ) and D ( V . ) . PROOF .Passing without loss of generality from t to rth , we may assume that 0 € 0 ( V ) .

Let Vi = Vm , where we choose m so large that 2pm 2 [ n / 2 ] + 1 . Then , by ...

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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