## Linear Operators: Spectral theory |

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Page 861

Clearly if x - 1

' ) ] = 4 , ( T ; ' ) = = T , ' ( ga ) , and if a = T le , then az = T ; lz for every z e X . Also

xa = Tya = e = T ; ? ( T2e ) = T ( ex ) = ( T ; e ) x = ax . Thus x - 1

Clearly if x - 1

**exists**then Tr - Tc = T & T2 - 1 = 1 . If Til**exists**in B ( x ) , then TE ( T' ) ] = 4 , ( T ; ' ) = = T , ' ( ga ) , and if a = T le , then az = T ; lz for every z e X . Also

xa = Tya = e = T ; ? ( T2e ) = T ( ex ) = ( T ; e ) x = ax . Thus x - 1

**exists**and Trx ...Page 1057

Thus ( 2 ) gives j→ Jen y " ^ 27 F ( K * f ) ( u ) = ( 27 ) - 1 / 2 lim PL 1197 x ( y ) { |

ciua f ( – y ) dx ) dy = { uim a la supervdy ) Fury ) , provided only that the limit in

the braces in this last equation

...

Thus ( 2 ) gives j→ Jen y " ^ 27 F ( K * f ) ( u ) = ( 27 ) - 1 / 2 lim PL 1197 x ( y ) { |

ciua f ( – y ) dx ) dy = { uim a la supervdy ) Fury ) , provided only that the limit in

the braces in this last equation

**exists**. Thus , to complete the proof of the present...

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

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