## Linear Operators: Spectral theory |

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Page 922

T2, ..., are defined. 1 LEMMA. Let S, T, S,, T., n > 1 be bounded linear operators in

Hilbert space with S, -> S, T, -> T in the strong operator

**topology**, i.e., T., a -> Tr for every a in the space upon which the operators T, T1,T2, ..., are defined. 1 LEMMA. Let S, T, S,, T., n > 1 be bounded linear operators in

Hilbert space with S, -> S, T, -> T in the strong operator

**topology**. Then S.--To ...Page 1420

Conversely, let {f,} converge to zero in the

+|Ti(t-Ht')f, -> 0. If {f} is not bounded in Q(TA(r)), there is a subsequence {f...} such

that h, → f, Ti(t)f, converges to zero in Ś, and is bounded in Q(T(r)). By hypothesis

...

Conversely, let {f,} converge to zero in the

**topology**of $(Ti(t-i-t')), that is, let [+] fal+|Ti(t-Ht')f, -> 0. If {f} is not bounded in Q(TA(r)), there is a subsequence {f...} such

that h, → f, Ti(t)f, converges to zero in Ś, and is bounded in Q(T(r)). By hypothesis

...

Page 1921

(See Operator

, (419) study of, I.6 norm or strong, in a ... I.4–8

(49)

(See Operator

**topology**) metric, definition, I.6.1 (18) metric or strong, in a B-space, (419) study of, I.6 norm or strong, in a ... I.4–8

**topological**group, definition, II.1.1(49)

**topological**space, definition, I.4.1 (10) study of, I.4–8 weak, in a B-space, ...### 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