## 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 ( { uis } ) . We observe that by Lemma 7 , the ...

An

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

**elements**of L ? ( { u is } ) modulo { u is } -nullfunctions will be denoted by L2 ( { uis } ) . We observe that by Lemma 7 , the ...

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