## Linear Operators: Spectral theory |

### From inside the book

Results 1-3 of 85

Page 1459

corollary follows from Corollary 7 and Definition 25(b). Q.E.D. 31 CoRollARY.

Suppose in addition to the hypotheses of Theorem 8 that the coefficients p, are

real ...

**PROOF**. It is obvious from Definition 20 that t is bounded below. Thus the presentcorollary follows from Corollary 7 and Definition 25(b). Q.E.D. 31 CoRollARY.

Suppose in addition to the hypotheses of Theorem 8 that the coefficients p, are

real ...

Page 1724

= (f, Sg) for f in Q(T) and g in Q(S). ... It follows as in the

Corollary 11 that there exist bounded operators A and B mapping H'."(I) into itself

...

**PRoof**. By the preceding lemma and by Corollary 11 it suffices to show that (Tf, g)= (f, Sg) for f in Q(T) and g in Q(S). ... It follows as in the

**proof**of formula (6) ofCorollary 11 that there exist bounded operators A and B mapping H'."(I) into itself

...

Page 1750

We shall see, however, that this fact is needed in the course of the

Theorem 1, and shall prove it by a direct method where it is needed. Remark 2.

The theorem is false if no boundedness restriction is imposed on the coefficient

matrices ...

We shall see, however, that this fact is needed in the course of the

**proof**ofTheorem 1, and shall prove it by a direct method where it is needed. Remark 2.

The theorem is false if no boundedness restriction is imposed on the coefficient

matrices ...

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### 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