## Linear Operators: Spectral theory |

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Page 1420

... Q(TA(r)). (b) Let A be any bounded subset in 3)(Ti(r)). If A is considered as a

subset of $), then the restriction of Ti(t') to A is a continuous mapping of A into $5.

Then, assuming that t+t' has a non-

...

... Q(TA(r)). (b) Let A be any bounded subset in 3)(Ti(r)). If A is considered as a

subset of $), then the restriction of Ti(t') to A is a continuous mapping of A into $5.

Then, assuming that t+t' has a non-

**zero**leading coefficient, (A) the Hilbert spaces...

Page 1432

Suppose first that the end point under consideration is finite so that without loss of

generality we can suppose it to be at

the leading coefficient a, of r, we can write the equation (r—A)f = 0 in the form * ...

Suppose first that the end point under consideration is finite so that without loss of

generality we can suppose it to be at

**zero**. Then, dividing through if necessary bythe leading coefficient a, of r, we can write the equation (r—A)f = 0 in the form * ...

Page 1463

Since all the terms in the integral on the right are non-negative, we must have fif,

fif, identically

fif," is constant. Moreover, since f, and f, have only a finite number of

Since all the terms in the integral on the right are non-negative, we must have fif,

fif, identically

**zero**in [c,d]. Thus (fifi') = f°(fifs—ff.) is identically**zero**in sc, d, so thatfif," is constant. Moreover, since f, and f, have only a finite number of

**zeros**in [c ...### What people are saying - Write a review

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

34 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 Akad algebra Amer analytic applied assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complete 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 measure multiplicity neighborhood norm obtained partial positive preceding present problem projection PRoof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero