## Linear Operators: Spectral theory |

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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 70 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 ( Tx , x ) > 0 forevery x 70 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 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 , 00 ) . PROOF . Let Ebe the ...

Page 1338

Let { Mis } be a

respect to a

defined by the equations Mijle ) S.am mij ( 2 ) u ( da ) , where e is any bounded

Borel ...

Let { Mis } be a

**positive**matrix measure whose elements Mis are continuous withrespect to a

**positive**o - finite measure kl . If the matrix of densities { mi ; } isdefined by the equations Mijle ) S.am mij ( 2 ) u ( da ) , where e is any bounded

Borel ...

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### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Copyright | |

57 other sections not shown

### Common terms and phrases

additive Akad algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure multiplicity 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