## Linear Operators: Spectral theory |

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Page 1247

Q.E.D. The next lemma shows that a positive self adjoint transformation has a

transformation, there is a

* = T. PRoof.

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 adjointtransformation, there is a

**unique**positive self adjoint transformation A such that A* = T. PRoof.

Page 1250

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

Lemma ... Since A is

P(Air) = Tr. Further the extension of P by continuity from SR(A) - - to R(A) is

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

**unique**. ByLemma ... Since A is

**unique**, P is**uniquely**determined on R(A) by the equation ofP(Air) = Tr. Further the extension of P by continuity from SR(A) - - to R(A) is

**unique**.Page 1378

matria, measure {6,3, i, j = 1,..., k of Theorem 23 is

p, - 0, if i > k or j > k. PRoof. Suppose that ol, ..., o, is a determining set for T. Then

it is evident from Theorem 23 that if we define {p,}, i, j = 1,..., n, by p,(e) = 6,06), i, ...

matria, measure {6,3, i, j = 1,..., k of Theorem 23 is

**unique**, and 6, = p, i, j = 1,..., k;p, - 0, if i > k or j > k. PRoof. Suppose that ol, ..., o, is a determining set for T. Then

it is evident from Theorem 23 that if we define {p,}, i, j = 1,..., n, by p,(e) = 6,06), i, ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

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### Other editions - View all

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

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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 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