## 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 # 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 ( Tx , 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 , 00 ) . Proof . Let E bethe ...

Page 1338

Let { M is } be a

respect to a

defined by the equations Misle ) = S . m . , ( 2 ) u ( da ) , where e is any bounded

Borel ...

Let { M is } be a

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

**positive**o - finite measure u . If the matrix of densities { mis } isdefined by the equations Misle ) = S . m . , ( 2 ) u ( da ) , where e is any bounded

Borel ...

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### Contents

IX | 859 |

extensive presentation of applications of the spectral theorem | 911 |

Miscellaneous Applications | 937 |

Copyright | |

20 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f give given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero