## Linear Operators: Spectral theory |

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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 s (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

Notational Conventions and Preliminaries Throughout the rest of the present

chapter , the symbol J will

integers . We write | J \ = k , min J = min , siskli max J = maxisiskli . It will be

convenient ...

Notational Conventions and Preliminaries Throughout the rest of the present

chapter , the symbol J will

**denote**an index , i . e . , a k - tuple J = [ j1 , . . . , jx ] ofintegers . We write | J \ = k , min J = min , siskli max J = maxisiskli . It will be

convenient ...

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

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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additive Akad 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 singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero