## Linear Operators: Spectral theory |

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Page 877

Then an

Consequently the spectrum of y as an

an

...

Then an

**element**y in Y has an inverse in X if and only if it has an inverse in Y .Consequently the spectrum of y as an

**element**of y is the same as its spectrum asan

**element**of X . Proof . If y - l exists as an**element**of Y then , since X and Y have...

Page 878

Clearly the requirement that x and g ( u ) = u be corresponding

determines the * - isomorphism uniquely and we are thus led to the following

definition . 12 DEFINITION . Let x be an

and let fe C ...

Clearly the requirement that x and g ( u ) = u be corresponding

**elements**determines the * - isomorphism uniquely and we are thus led to the following

definition . 12 DEFINITION . Let x be an

**element**of a commutative B * - algebraand let fe C ...

Page 1339

An

The set of all equivalence classes of

functions will be denoted by L2 ( { Mis } ) . We observe that by Lemma 7 , the ...

An

**element**F of Ly ( { uis } ) will be said to be a { Mis } - null function if ( F ] = 0 .The set of all equivalence classes of

**elements**of Ly ( { uis } ) modulo { Miš - nullfunctions will be denoted by L2 ( { Mis } ) . We observe that by Lemma 7 , the ...

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### Contents

IX | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Compact Groups | 945 |

Copyright | |

46 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 adjoint operator algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently consider constant contains continuous converges Corollary corresponding defined Definition denote dense derivatives determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function 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 satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero