## Linear Operators: Spectral theory |

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Page 1051

Statement (i) is

consequences of Definition 1 and of the formulae J.W.G.)l = s.40a)do, s.v.G.)are a

"s." ()ār, which are valid for every Lebesgue integrable function p. Statement (iv)

will ...

Statement (i) is

**evident**from Definition 1. Statements (ii) and (iii) are**evident**consequences of Definition 1 and of the formulae J.W.G.)l = s.40a)do, s.v.G.)are a

"s." ()ār, which are valid for every Lebesgue integrable function p. Statement (iv)

will ...

Page 1347

This procedure has the

of the complex variable %; but it has drawbacks which, though less

nevertheless decisive. Suppose, for example, that we study the self adjoint ...

This procedure has the

**evident**advantage that it makes o,(., A) an entire functionof the complex variable %; but it has drawbacks which, though less

**evident**, arenevertheless decisive. Suppose, for example, that we study the self adjoint ...

Page 1631

It is

closed graph theorem (II.2.4), T is continuous. Hence given any e > 0 and any k,

there is an integer l and a 6 × 0 such that u(k, Togo, . . . g., 1]) < e provided that |s,.

It is

**evident**that T is linear, and equally**evident**that T is closed. Hence, by theclosed graph theorem (II.2.4), T is continuous. Hence given any e > 0 and any k,

there is an integer l and a 6 × 0 such that u(k, Togo, . . . g., 1]) < e provided that |s,.

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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