## Linear Operators: Spectral theory |

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Page 907

negative real axis respectively . Proof . If N is a bounded normal operator then ,

by Corollary IX.3.15 , NN * = N * N = 1 if and only if 2ī = 1 for every spectral point ...

**positive**if and only if its spectrum lies on the unit circle , the real axis , or the non -negative real axis respectively . Proof . If N is a bounded normal operator then ,

by Corollary IX.3.15 , NN * = N * N = 1 if and only if 2ī = 1 for every spectral point ...

Page 1247

Q.E.D. Next we shall require some information on

transformations and their square roots . 2 LEMMA . A self adjoint transformation T

is

the ...

Q.E.D. Next we shall require some information on

**positive**self adjointtransformations and their square roots . 2 LEMMA . A self adjoint transformation T

is

**positive**if and only if o ( T ) is a subset of the interval [ 0 , 0 ) . PROOF . Let E bethe ...

Page 1338

( ii ) we have MylŪem ) = { Huslem ) m = 1 me1 for each sequence of disjoint

Borel sets with bounded union . 7 LEMMA . Let { uis } be a

measure whose elements Mis are continuous with respect to a

measure u .

( ii ) we have MylŪem ) = { Huslem ) m = 1 me1 for each sequence of disjoint

Borel sets with bounded union . 7 LEMMA . Let { uis } be a

**positive**matrixmeasure whose elements Mis are continuous with respect to a

**positive**o - finitemeasure u .

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