## Linear Operators: Spectral theory |

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Page 906

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

and if ( Tx , x ) 2 0 for every x in v ; and

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

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

**positive**if it is self adjointand if ( Tx , x ) 2 0 for every x in v ; and

**positive**definite if it is**positive**and ( Tx , x )> 0 for every 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

be ...

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 , 00 ) . Proof . Let Ebe ...

Page 1338

( ii ) we have Mis ( Uem ) = { Hilem ) m = 1 mel for each sequence of disjoint Borel

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

whose elements Mij are continuous with respect to a

.

( ii ) we have Mis ( Uem ) = { Hilem ) m = 1 mel for each sequence of disjoint Borel

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

**positive**matrix measurewhose elements Mij are continuous with respect to a

**positive**o - finite measure u.

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### Contents

BAlgebras | 861 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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additive Akad algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complete Consequently 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 Nauk neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero