## Linear Operators: Spectral theory |

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Page 926

25 Show that the bounded operator T in Hilbert space is compact if and only if

either ( a ) Txn 0 strongly whenever xn → 0

whenever Xm → ( )

that ( Txn ...

25 Show that the bounded operator T in Hilbert space is compact if and only if

either ( a ) Txn 0 strongly whenever xn → 0

**weakly**, or ( b ) ( Txn , xn ) → ( )whenever Xm → ( )

**weakly**. ( Hint : For ( b ) , show that the hypothesis impliesthat ( Txn ...

Page 932

Let S be an abstract set and a field ( resp . o - field ) of subsets of S . Let F be an

additive ( resp .

operators on a Hilbert space H satisfying F ( $ ) = 0 and F ( S ) = I . Then there

exists ...

Let S be an abstract set and a field ( resp . o - field ) of subsets of S . Let F be an

additive ( resp .

**weakly**countably additive ) function on { to the set of positiveoperators on a Hilbert space H satisfying F ( $ ) = 0 and F ( S ) = I . Then there

exists ...

Page 1442

Since every bounded sequence of functions in Hilbert space contains a

convergent subsequence , we may assume without loss of generality that { m } is

Since every bounded sequence of functions in Hilbert space contains a

**weakly**convergent subsequence , we may assume without loss of generality that { m } is

**weakly**convergent . Then by Lemma 2 . 16 , { m } converges**weakly**in the ...### What people are saying - Write a review

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

Copyright | |

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