## Linear Operators: Spectral theory |

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Page 1154

Since the product group R ( 2 ) = RX R is locally compact and o - compact , it has

a Haar measure 2 ( 2 ) defined on its Borel field { ( 2 ) and what we shall prove is

that for some

Since the product group R ( 2 ) = RX R is locally compact and o - compact , it has

a Haar measure 2 ( 2 ) defined on its Borel field { ( 2 ) and what we shall prove is

that for some

**constant**c , ( R ( 2 ) , 3 ( 2 ) , 2 ( 2 ) ) = c ( R , E , 2 ) x ( R , E , 2 ) .Page 1176

Subtracting a suitable

suppose without loss of generality that kn ... here we have used the uniform

boundedness of the functions kn and of their variations to conclude that the

Subtracting a suitable

**constant**on from each of the functions kn , we maysuppose without loss of generality that kn ... here we have used the uniform

boundedness of the functions kn and of their variations to conclude that the

**constants**on are ...Page 1730

Moreover , there exists a

C ) . Now we shall prove an important lemma on elliptic partial differential

equations with

differential ...

Moreover , there exists a

**constant**A < o such that l ( if , g ) | < Altliglo , f , ge C . . (C ) . Now we shall prove an important lemma on elliptic partial differential

equations with

**constant**coefficients . 18 LEMMA . Let o be a formal partialdifferential ...

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### Contents

IX | 859 |

Eigenvalues and Eigenvectors | 903 |

Spectral Representation | 911 |

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 Russian satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero