## Linear Operators: Spectral theory |

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Page 1191

However, this operator is not self

that any function g with a continuous first derivative has the property that (#1 -(, ; *

(i. 'd *)- hijo, fe i}). and thus any such g, even though it fails to vanish at one of the

...

However, this operator is not self

**adjoint**for it is clear from the above equationsthat any function g with a continuous first derivative has the property that (#1 -(, ; *

(i. 'd *)- hijo, fe i}). and thus any such g, even though it fails to vanish at one of the

...

Page 1247

Q.E.D. Next we shall require some information on positive self

transformations and their square roots. 2 LEMMA. A self

is positive if and only if o(T) is a subset of the interval [0, 00). PRoof. Let E be the

resolution ...

Q.E.D. Next we shall require some information on positive self

**adjoint**transformations and their square roots. 2 LEMMA. A self

**adjoint**transformation Tis positive if and only if o(T) is a subset of the interval [0, 00). PRoof. Let E be the

resolution ...

Page 1290

that since to ti - (T211)*, the operator X (–) () to () is formally self

only that the coefficients p, are real. In the same way, the formal differential

operator (i/2)(d/dt)"{p(t)(d/dt)+(d/dt)p(t)}(d/dt)" is formally self

p(t) is ...

that since to ti - (T211)*, the operator X (–) () to () is formally self

**adjoint**providedonly that the coefficients p, are real. In the same way, the formal differential

operator (i/2)(d/dt)"{p(t)(d/dt)+(d/dt)p(t)}(d/dt)" is formally self

**adjoint**provided thatp(t) is ...

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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