## Linear Operators: Spectral theory |

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Page 1420

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

3 (T1(t +t')) and 3)(Ti(r)) have the same elements and equivalent topologies; (B)

the differential operators t and t' have the same deficiency indices. PRoof.

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

**zero**leading coefficient, (A) the Hilbert spaces3 (T1(t +t')) and 3)(Ti(r)) have the same elements and equivalent topologies; (B)

the differential operators t and t' have the same deficiency indices. PRoof.

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 (1–2)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 (1–2)f = 0 in the form ...

Page 1463

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

f. f. identically

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

c, ...

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

f. f. identically

**zero**in [c, d). Thus (fif, ')' = f°(fif, f. f.) is identically**zero**in [c,d], sothat fif," is constant. Moreover, since f, and f, have only a finite number of

**zeros**in [c, ...

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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