## Linear Operators, Part 2 |

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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

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

( ii ) we have Misc Üem ) = Erslem ) m = 1 mel for each sequence of disjoint Borel

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

whose elements Mis are continuous with respect to a

.

( ii ) we have Misc Üem ) = Erslem ) m = 1 mel for each sequence of disjoint Borel

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

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

**positive**o - finite measure u.

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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additive adjoint operator 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 derivatives determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function give given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure multiplicity neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero