## Linear Operators: Spectral theory |

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Page 1142

The validity of the

from its validity in the range 2 < p < oc and from Lemma 9.14. Q.E.D. In what

follows, we will use the symbols p and m to denote the continuous extension to

the ...

The validity of the

**present**theorem in the range 1 < p < 2 now follows at oncefrom its validity in the range 2 < p < oc and from Lemma 9.14. Q.E.D. In what

follows, we will use the symbols p and m to denote the continuous extension to

the ...

Page 1675

By what has been shown in the first paragraph of the

that 6, F is in Ho (C). This completes the proof of the direct part of (i) of the

lemma. To prove the converse, let F be in Ho (C) and let 8, F be in Ho (C).

By what has been shown in the first paragraph of the

**present**proof, F = 6, F, sothat 6, F is in Ho (C). This completes the proof of the direct part of (i) of the

**present**lemma. To prove the converse, let F be in Ho (C) and let 8, F be in Ho (C).

Page 1703

In the

partial differential operators to be defined below. A crucial theorem in the

development of the theory of Chapter XIII was Theorem XIII.2.10, which was

based on ...

In the

**present**section it will be seen that it can, at least for the class of ellipticpartial differential operators to be defined below. A crucial theorem in the

development of the theory of Chapter XIII was Theorem XIII.2.10, which was

based on ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

48 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 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