## 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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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