## Linear Operators: Spectral theory |

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Page 1000

If {fi} were known to be uniformly convergent in a neighborhood of U, the

analyticity of its limit fo would be

sequence f, is uniformly convergent on any region containing an interval of the

real axis and so ...

If {fi} were known to be uniformly convergent in a neighborhood of U, the

analyticity of its limit fo would be

**clear**. Unfortunately it is not**clear**that thesequence f, is uniformly convergent on any region containing an interval of the

real axis and so ...

Page 1243

According to VIII.1.8 and VIII.1.11, A is a closed operator with dense domain

whose resolvent set includes the whole half plane 3*(A) > 0. Moreover, we have

[+] R(A; A)a = | e-*U(t)ardt, 3(A) > 0. Put S(t) = U(t)* = {U(t)}-1 for t > 0. It is

that ...

According to VIII.1.8 and VIII.1.11, A is a closed operator with dense domain

whose resolvent set includes the whole half plane 3*(A) > 0. Moreover, we have

[+] R(A; A)a = | e-*U(t)ardt, 3(A) > 0. Put S(t) = U(t)* = {U(t)}-1 for t > 0. It is

**clear**that ...

Page 1689

Indeed, if {f,} is a Cauchy sequence in L.(I), it is

sequence in L(I) for |J| < k, so that there exist functions g, g' in L,(I) such that lim, ...

If...—g, =0 and limm-old'f,-g", = 0. It is then

Indeed, if {f,} is a Cauchy sequence in L.(I), it is

**clear**from (i) that {6'f,} is a Cauchysequence in L(I) for |J| < k, so that there exist functions g, g' in L,(I) such that lim, ...

If...—g, =0 and limm-old'f,-g", = 0. It is then

**clear**from Definition 3.26 that limn.### 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