## Linear operators. 2. Spectral theory : self adjoint operators in Hilbert Space |

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Page 970

If Xe denotes the characteristic function of the set e in E , and if f is in Ly ( R ) , then Xef is in Ly ( R ) L2 ( R ) and f is the

If Xe denotes the characteristic function of the set e in E , and if f is in Ly ( R ) , then Xef is in Ly ( R ) L2 ( R ) and f is the

**limit**in the norm of L ( R ) of the generalized sequence { xef } . Hence , by Theorem 9 , tf is the ...Page 1124

If E ,, E are in F and q ( En ) increases to the

If E ,, E are in F and q ( En ) increases to the

**limit**9 ( E ) , then it follows from what we have already proved that E is an increasing sequence of projections and E , SE . If Ec is the strong**limit**of En , then E. SE and Q ( E ) = Q ...Page 1129

1 sn sm . m Thus E ( e ) is the strong

1 sn sm . m Thus E ( e ) is the strong

**limit**of the operators Om . On the other hand , it follows from Theorem X.2.1 that Om belongs to the algebra A , so that one is a**limit**of linear combinations of products of the operators E. Since ...### What people are saying - Write a review

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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### Other editions - View all

Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert ... Nelson Dunford,Jacob T. Schwartz No preview available - 1988 |

Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert ... Nelson Dunford,Jacob T. Schwartz No preview available - 1988 |

Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert ... Nelson Dunford,Jacob T. Schwartz No preview available - 1988 |

### Common terms and phrases

additive adjoint operator 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 neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero