## Linear Operators: Spectral theory |

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Page 861

1 = 1 . If T . '

if a = Tate , then az = 1 ; 12 for every z e X . Also xa = Tya = e = 7 : " ( T . e ) = 1 ; ?

( ex ) = ( T74e ) x = ax . Thus 2 - 1

1 = 1 . If T . '

**exists**in B ( X ) , then Te [ ( T y ) 2 ] = y2 , ( TÖ ' y ) 2 = T ? ? ( yz ) , andif a = Tate , then az = 1 ; 12 for every z e X . Also xa = Tya = e = 7 : " ( T . e ) = 1 ; ?

( ex ) = ( T74e ) x = ax . Thus 2 - 1

**exists**and Tol % = x - 12 . 2 DEFINITION .Page 1057

F ( K * f ) ( u ) = ( 27 ) - n / 2 Jim P j = Jen y " af 2 ( y ) = lim P litoo den lylna

provided only that the limit in the braces in this last equation

complete the proof of the present lemma , it suffices to show that ( 3 ) ( u ) = P | 9

pivu dy ...

F ( K * f ) ( u ) = ( 27 ) - n / 2 Jim P j = Jen y " af 2 ( y ) = lim P litoo den lylna

provided only that the limit in the braces in this last equation

**exists**. Thus , tocomplete the proof of the present lemma , it suffices to show that ( 3 ) ( u ) = P | 9

pivu dy ...

Page 1262

Then there

such that Ax = PQx , XEH , P denoting the orthogonal projection of Hi on H . 29

Let { Tn } be a sequence of bounded operators in Hilbert space H . Then there

Then there

**exists**a Hilbert space H , 2H , and an orthogonal projection Q in H ,such that Ax = PQx , XEH , P denoting the orthogonal projection of Hi on H . 29

Let { Tn } be a sequence of bounded operators in Hilbert space H . Then there

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### Contents

IX | 859 |

extensive presentation of applications of the spectral theorem | 911 |

Miscellaneous Applications | 937 |

Copyright | |

20 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 |

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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 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