## Linear Operators: Spectral operators |

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Page 2149

Condition (A) of Section 2 is satisfied if the spectrum of T is nowhere

complex plane. Proof. If the resolvent set is

continuous, extensions of Jf(A; T)x must coincide on their common domain of ...

Condition (A) of Section 2 is satisfied if the spectrum of T is nowhere

**dense**in thecomplex plane. Proof. If the resolvent set is

**dense**, then any two analytic, or evencontinuous, extensions of Jf(A; T)x must coincide on their common domain of ...

Page 2156

are both

- T)Nx = 0} is

"x=0} is also

are both

**dense**in X. Since 9W2 is**dense**in X, the manifold (XJ - T)"9Jl2 + {a; | (XJ- T)Nx = 0} is

**dense**in X, so that (XJ -T)"(X2I- T)NX + {x | (XJ -T)"x = 0} + {x\(XaI-T)"x=0} is also

**dense**in X. By Lemma 7, o(x) c y if (A, / — T)"x = 0 for i = 1 or i = 2, ...Page 2159

The union of all intervals of constancy relative to T is an open set

Proof. It is clear that the union of intervals of constancy is open. To see that it is

y, ...

The union of all intervals of constancy relative to T is an open set

**dense**in ro .Proof. It is clear that the union of intervals of constancy is open. To see that it is

**dense**, let y be a closed subarc of ro having positive length and let y„ = {A0 1 A0 6y, ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

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adjoint operator Akad Amer analytic applications arbitrary assume B-space Banach space belongs Boolean algebra Borel sets bounded bounded operator Chapter clear closed commuting compact complex consider constant contains continuous converges Corollary corresponding countably additive defined Definition denote dense determined differential operator domain elements equation equivalent established example exists extension fact finite follows formula function given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator Math Moreover multiplicity Nauk norm normal perturbation plane positive preceding present problem projections Proof properties prove range resolution resolvent restriction Russian satisfies scalar type seen sequence shown shows similar solution spectral measure spectral operator spectrum subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector weakly zero