## Linear Operators: Spectral theory |

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Page 1247

... o ( Tn ) C [ 0 , 00 ) . Thus [ * ] shows that o ( T ) C [ 0 , 00 ) . Q.E.D. The next

lemma shows that a positive self adjoint transformation has a

square root ” . 3 LEMMA . If T is a positive self adjoint transformation , there is a

... o ( Tn ) C [ 0 , 00 ) . Thus [ * ] shows that o ( T ) C [ 0 , 00 ) . Q.E.D. The next

lemma shows that a positive self adjoint transformation has a

**unique**positive “square root ” . 3 LEMMA . If T is a positive self adjoint transformation , there is a

**unique**...Page 1250

Since A is

= Tx . Further the extension of P by continuity from R ( A ) to R ( A ) is

Since P is zero on R ( A ) ' it follows that P is

8.

Since A is

**unique**, P is**uniquely**determined on R ( A ) by the equation of P ( Ax )= Tx . Further the extension of P by continuity from R ( A ) to R ( A ) is

**unique**.Since P is zero on R ( A ) ' it follows that P is

**uniquely**determined by T. Q.E.D. 0 .8.

Page 1513

Let F. be the

and exponents which has the form 2-2 ( 1 + + ... ) near z = 0. Then , since Fa and

Ft together comprise a basis for the solutions of our equation , we have a ...

Let F. be the

**unique**solution of the equation with these same regular singularitiesand exponents which has the form 2-2 ( 1 + + ... ) near z = 0. Then , since Fa and

Ft together comprise a basis for the solutions of our equation , we have a ...

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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

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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 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