## Linear operators: Spectral operators |

### From inside the book

Results 1-3 of 86

Page 2257

It follows by Definition VII.9.3 that E(o; T)=E(t(o); R) for each

a of T. Moreover, it is clear that as a runs over the family K of all

subsets of a(T), t(o) runs over the family of all

which ...

It follows by Definition VII.9.3 that E(o; T)=E(t(o); R) for each

**compact**spectral seta of T. Moreover, it is clear that as a runs over the family K of all

**compact**opensubsets of a(T), t(o) runs over the family of all

**compact**open subsets of a{R)which ...

Page 2360

It will be shown below that \TvR(fi; T)A\ ^ £ for fi. in F, and t sufficiently large. From

this it will then follow as above that the function f(p.) = R(fi; T + P)f is uniformly

bounded. It will also be shown that T'y is

.

It will be shown below that \TvR(fi; T)A\ ^ £ for fi. in F, and t sufficiently large. From

this it will then follow as above that the function f(p.) = R(fi; T + P)f is uniformly

bounded. It will also be shown that T'y is

**compact**. From this, (iii), and Theorem VI.

Page 2462

is

by Corollary VI. 5. 5, and thus proof of Corollary 11 is complete. Q.E.D. 12 Lemma

. If C is a

is

**compact**. Put C — QBlt and D = R2 , so that V = CD. The operator C is**compact**by Corollary VI. 5. 5, and thus proof of Corollary 11 is complete. Q.E.D. 12 Lemma

. If C is a

**compact**operator in §, and {Tn} is a uniformly bounded sequence of ...### What people are saying - Write a review

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

### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

47 other sections not shown

### Other editions - View all

### Common terms and phrases

adjoint operator algebra of projections Amer analytic arbitrary asymptotic B-space Banach space Boolean algebra Borel sets boundary conditions bounded Borel function bounded linear operator bounded operator commuting compact complex numbers complex plane constant contains continuous functions converges Corollary countably additive Definition denote dense differential operator disjoint Doklady Akad domain eigenvalues elements equation exists finite number Foias follows from Lemma follows from Theorem formal differential operator formula Hence Hilbert space hypothesis identity inequality inverse Lebesgue Math matrix measurable functions multiplicity Nauk SSSR norm normal operators operators in Hilbert perturbation polynomial Proof properties prove quasi-nilpotent restriction Russian satisfies scalar operator scalar type operator scalar type spectral Section semi-group sequence shows spectral measure spectral operator spectral theory spectrum strong operator topology subset Suppose trace class type spectral operator unbounded uniformly bounded unique vector weakly complete zero