## Linear Operators: Spectral theory |

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Page 879

It is clear that the smallest closed subalgebra of B ( H ) which

operator T , its adjoint T * , and the identity I is a commutative B * -algebra . Thus

we may state the following corollary . 15 COROLLARY . Let T be a normal ...

It is clear that the smallest closed subalgebra of B ( H ) which

**contains**a normaloperator T , its adjoint T * , and the identity I is a commutative B * -algebra . Thus

we may state the following corollary . 15 COROLLARY . Let T be a normal ...

Page 995

If y and f are in L. ( R ) and L ( R ) respectively and if f ( m ) O for every m in the

spectral set o ( Q ) , then off * 9 )

make an indirect proof by supposing that mo is an isolated point of olf * 9 ) . Since

olf ...

If y and f are in L. ( R ) and L ( R ) respectively and if f ( m ) O for every m in the

spectral set o ( Q ) , then off * 9 )

**contains**no isolated points . PROOF . We shallmake an indirect proof by supposing that mo is an isolated point of olf * 9 ) . Since

olf ...

Page 996

From Lemma 12 ( b ) it is seen that olf * 9 ) Cole ) and from Lemma 12 ( c ) and

the equation of = Tf it follows that o ( f * Q )

Hence o ( f * 9 ) is a closed subset of the boundary of o ( q ) . Since f * q = 0 it

follows ...

From Lemma 12 ( b ) it is seen that olf * 9 ) Cole ) and from Lemma 12 ( c ) and

the equation of = Tf it follows that o ( f * Q )

**contains**no interior point of o ( 9 ) .Hence o ( f * 9 ) is a closed subset of the boundary of o ( q ) . Since f * q = 0 it

follows ...

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