## Linear Operators: Spectral theory |

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Page 1187

Just as in the case of a

is defined to be the set of all complex numbers a such that ( 21 – T ) - 1 exists as

an everywhere defined

Just as in the case of a

**bounded operator**the resolvent set p ( T ) of an operator Tis defined to be the set of all complex numbers a such that ( 21 – T ) - 1 exists as

an everywhere defined

**bounded operator**. For 2 in p ( T ) the symbol R ( 2 ; T ) ...Page 1190

An

dense so that T * is defined then the notion of symmetry is equivalent to the

inclusion T * 2 T . Of course if T is a

the ...

An

**operator**T may be symmetric without having a dense domain but if D ( T ) isdense so that T * is defined then the notion of symmetry is equivalent to the

inclusion T * 2 T . Of course if T is a

**bounded**everywhere defined**operator**thenthe ...

Page 1273

Weyl [ 5 ] showed this to be the case for differential

that the inverse

set y ...

Weyl [ 5 ] showed this to be the case for differential

**operators**. If T is a linear**operator**with dense domain , let y ( T ) be the set of all complex numbers à suchthat the inverse

**operator**( T - 21 ) - 1 exists and is**bounded**on its domain . Theset y ...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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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 singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero