## Linear Operators: Spectral theory |

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Page 861

In this connection it is desirable to note that an element T. et(3) has an inverse as

an element of B(3) if and only if a has an inverse in 3 and that when this inverse T

.”

In this connection it is desirable to note that an element T. et(3) has an inverse as

an element of B(3) if and only if a has an inverse in 3 and that when this inverse T

.”

**exists**, then T.” = T-1. Clearly if a-1**exists**then T-1T = T, T-1 = I. If T.'**exists**in ...Page 1057

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

complete the proof of the present lemma, it suffices to show that Q Q (3) 6(u) = %

o e” dy= lim s o e” dy e” ly ;:.." esswish |y|

integral ...

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 Q Q (3) 6(u) = %

o e” dy= lim s o e” dy e” ly ;:.." esswish |y|

**exists**for each u. By Lemma 2, theintegral ...

Page 1261

23 If an operator T has a closed linear extension there

linear extension T such that if T is any closed linear extension of T then TC T1. T

is called the closure of T. (a) There

closed ...

23 If an operator T has a closed linear extension there

**exists**a unique closedlinear extension T such that if T is any closed linear extension of T then TC T1. T

is called the closure of T. (a) There

**exists**a densely defined operator with noclosed ...

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

34 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 |

### Common terms and phrases

additive Akad algebra Amer analytic applied assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complete 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 measure multiplicity neighborhood norm obtained partial positive preceding present problem projection PRoof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero