## Linear operators. 2. Spectral theory : self adjoint operators in Hilbert Space |

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Page 906

... self adjoint , symmetric or Hermitian if T = T * ;

Tx , x ) 20 for every x in H ; and

every x + 0 in H. It is clear that all of these classes of operators are normal .

... self adjoint , symmetric or Hermitian if T = T * ;

**positive**if it is self adjoint and if (Tx , x ) 20 for every x in H ; and

**positive**definite if it is**positive**and ( Tr , x ) > 0 forevery x + 0 in H. It is clear that all of these classes of operators are normal .

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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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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