## Linear Operators: Spectral theory |

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Page 1223

In the theory of bounded operators, we have only to verify

T is everywhere defined and

situation is quite different. Consider, as an example, an operator which will be ...

In the theory of bounded operators, we have only to verify

**symmetry**(To DT), for ifT is everywhere defined and

**symmetric**, then To = T. But if T is unbounded thesituation is quite different. Consider, as an example, an operator which will be ...

Page 1236

A set of boundary conditions B, (a) = 0, i = 1,..., k, is said to be

equations B,(r) = B,(y) = 0, i = 1,..., k, imply the equation {a, y} = 0. 26 LEMMA. Let

T be an operator with finite deficiency indices. Every closed

of ...

A set of boundary conditions B, (a) = 0, i = 1,..., k, is said to be

**symmetric**if theequations B,(r) = B,(y) = 0, i = 1,..., k, imply the equation {a, y} = 0. 26 LEMMA. Let

T be an operator with finite deficiency indices. Every closed

**symmetric**extensionof ...

Page 1272

Marimal

then it has proper

are different from zero. A marimal

...

Marimal

**symmetric**operators. If T is a**symmetric**operator with dense domain,then it has proper

**symmetric**extensions provided both of its deficiency indicesare different from zero. A marimal

**symmetric**operator is one which has no proper...

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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