## Linear Operators: Spectral theory |

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Page 1247

If T is a positive self adjoint transformation , there is a

transformation A such that A2 = T . Proof . By Lemma 2 , o ( T ) C ( 0 , 0 ) and , by

Theorem 2 . 6 ( d ) , the positive function f ( a ) = at on o ( T ) defines the self ...

If T is a positive self adjoint transformation , there is a

**unique**positive self adjointtransformation A such that A2 = T . Proof . By Lemma 2 , o ( T ) C ( 0 , 0 ) and , by

Theorem 2 . 6 ( d ) , the positive function f ( a ) = at on o ( T ) defines the self ...

Page 1250

Finally we show that the decomposition T = PA of the theorem is

A is

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

Finally we show that the decomposition T = PA of the theorem is

**unique**. ... SinceA 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 ...Page 1632

On the other hand , if the Cauchy problem had a

, 81 of the prescribed data , then , by what has been proved above , the

phenomenon of local dependence would occur , and we would be able to

construct ...

On the other hand , if the Cauchy problem had a

**unique**solution for each pair 80, 81 of the prescribed data , then , by what has been proved above , the

phenomenon of local dependence would occur , and we would be able to

construct ...

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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