## Linear Operators: Spectral theory |

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Page 1000

If {f,} 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 {f,} 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 1393

It is

interval I, then the essential spectrum of the closed operator Ti(t) in L2(I) is called

the essential spectrum o.(1) of t. 2 LEMMA. Let 3: be a Banach space, and

suppose ...

It is

**clear**that o,(T) Co.(T). If t is a formal differential operator defined on theinterval I, then the essential spectrum of the closed operator Ti(t) in L2(I) is called

the essential spectrum o.(1) of t. 2 LEMMA. Let 3: be a Banach space, and

suppose ...

Page 1651

Thus G|I = F. If KCF = 0 and the function p in C. (I U Io) vanishes outside K, then it

is

= 0. This shows that Co. C Cr, and it is

Thus G|I = F. If KCF = 0 and the function p in C. (I U Io) vanishes outside K, then it

is

**clear**that popk vanishes outside a compact subset of I–Cr; thus G(p) = F(pkop)= 0. This shows that Co. C Cr, and it is

**clear**conversely that CF = Cal I C Co.### What people are saying - Write a review

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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