## 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 1.6(c), AP* = To. Hence T^T = AP*PA. Since, by Lemma 5, Po P is a

projection onto ot(A), it follows that T*T = A*. The uniqueness of A now follows

from ...

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

**unique**. ByLemma 1.6(c), AP* = To. Hence T^T = AP*PA. Since, by Lemma 5, Po P is a

projection onto ot(A), it follows that T*T = A*. The uniqueness of A now follows

from ...

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 |

Bounded Normal Operators in Hilbert Space | 887 |

Spectral Representation | 909 |

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