## Linear Operators: Spectral theory |

### From inside the book

Results 1-3 of 82

Page 1126

of the closed set C ; we shall

C ) . Since each projection in the spectral resolution of T and hence each

continuous function of T is a strong limit of linear combinations of the projections

Ei , it ...

of the closed set C ; we shall

**denote**this subspace of L2 [ 0 , 1 ] by the symbol L (C ) . Since each projection in the spectral resolution of T and hence each

continuous function of T is a strong limit of linear combinations of the projections

Ei , it ...

Page 1635

The symbol Er will

index for En if min J 21 and max J Sn . If & is in Un , so that 5 = ( $ 1 , . . . , 5m ] ,

and J ...

The symbol Er will

**denote**real Euclidean n - space , and the symbol U ” will**denote**complex unitary n - dimensional space . An index J will be said to be anindex for En if min J 21 and max J Sn . If & is in Un , so that 5 = ( $ 1 , . . . , 5m ] ,

and J ...

Page 1636

In general , unless the contrary is explicitly stated , J , ) , J , etc . , will

indices for E ” , that is , indices whose range of variation is restricted by the

condition min J 21 , max J Sn . The symbols J1 , I1 , , will similarly

for En + 1 ...

In general , unless the contrary is explicitly stated , J , ) , J , etc . , will

**denote**indices for E ” , that is , indices whose range of variation is restricted by the

condition min J 21 , max J Sn . The symbols J1 , I1 , , will similarly

**denote**indicesfor En + 1 ...

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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