## Linear Operators: Spectral theory |

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Page 1187

If B is a bounded

a, Br, I) is a Cauchy sequence in the

Br] in T(B). Thus the sequence {a,) converges to the point a in 3)(B) which proves

...

If B is a bounded

**closed**operator and if {a,) is a Cauchy sequence in 3)(B) then {[a, Br, I) is a Cauchy sequence in the

**closed**set T(B) and hence it has a limit sa,Br] in T(B). Thus the sequence {a,) converges to the point a in 3)(B) which proves

...

Page 1393

Let T be a

such that the range of AI —T is not

and is denoted by o,(T). It is clear that a.(T) Co.(T). If t is a formal differential

operator ...

Let T be a

**closed**operator in Hilbert space. Then the set of complex numbers Asuch that the range of AI —T is not

**closed**is called the essential spectrum of Tand is denoted by o,(T). It is clear that a.(T) Co.(T). If t is a formal differential

operator ...

Page 1436

By Corollary IV.8.2, every finite dimensional subspace of a B-space is

Thus, by the HahnBanach theorem (II.3.18) there exists a set af, ..., a of

continuous linear functionals on the B-space such that af(p) = 0 for 0 < i + j < k, a f

(q,) = 1.

By Corollary IV.8.2, every finite dimensional subspace of a B-space is

**closed**.Thus, by the HahnBanach theorem (II.3.18) there exists a set af, ..., a of

continuous linear functionals on the B-space such that af(p) = 0 for 0 < i + j < k, a f

(q,) = 1.

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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