## Linear Operators: Spectral theory |

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Page 1064

... subsequence , we may

clear that gm ( x ) = g ( x ) for all x . Hence q = g , proving that ( gl , 51 , 111 , if f is

in Co ( I ) and vanishes outside a bounded set . Next let | be in L ( E ” ) , and let {

Im } ...

... subsequence , we may

**assume**that gm ( x ) = Q ( x ) for almost all x . But it isclear that gm ( x ) = g ( x ) for all x . Hence q = g , proving that ( gl , 51 , 111 , if f is

in Co ( I ) and vanishes outside a bounded set . Next let | be in L ( E ” ) , and let {

Im } ...

Page 1120

Throughout the present section , we

Hilbert space is separable . Subdiagonal representations of an operator are

connected with the study of its invariant subspaces . Thus , the key to the situation

that we ...

Throughout the present section , we

**assume**for simplicity of statement thatHilbert space is separable . Subdiagonal representations of an operator are

connected with the study of its invariant subspaces . Thus , the key to the situation

that we ...

Page 1734

Since we have only to show that ff is in H ( k + 1 ) ( UI ) for some neighborhood U

CU , of p , it is clear that we may

70 = 1 . This will be

Since we have only to show that ff is in H ( k + 1 ) ( UI ) for some neighborhood U

CU , of p , it is clear that we may

**assume**without loss of generality that U2 = U ,70 = 1 . This will be

**assumed**in what follows . Making use of the properties ( i ) ...### What people are saying - Write a review

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

15 other sections not shown

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additive Akad algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure multiplicity neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result Russian satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero