## Linear Operators: Spectral operators |

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Page 2410

Then there exists a positive

that if \\A || < e(p, D) it follows that the operator T + <p(A) is similar to the operator

T and is therefore a spectral operator of scalar type. The following theorem ...

Then there exists a positive

**constant**e = e(p, D) depending only on p and D suchthat if \\A || < e(p, D) it follows that the operator T + <p(A) is similar to the operator

T and is therefore a spectral operator of scalar type. The following theorem ...

Page 2439

1.7-1 82f dkl dk, (ft) ^Me(l + \lc\n + 1y Let us now note that there exists a positive

absolute

, <o' e S. Indeed, if 6 is the angle between cd and to', it is clear that |tu — xco'\ > ...

1.7-1 82f dkl dk, (ft) ^Me(l + \lc\n + 1y Let us now note that there exists a positive

absolute

**constant**c such that (31) \cd — xw'\^c\<ju — ai'l for all 0 < a < 1 and all to, <o' e S. Indeed, if 6 is the angle between cd and to', it is clear that |tu — xco'\ > ...

Page 2441

... /(a) = 0(<x~{n-^) as a— >□ oo. This shows that there exists a finite absolute

we see that there exists a finite

\ + ...

... /(a) = 0(<x~{n-^) as a— >□ oo. This shows that there exists a finite absolute

**constant**c' such that 7(a)| ^ c'(l -f |a[)-n + 1. Using this inequality and using (37),we see that there exists a finite

**constant**M independent of e such that (43) \Vt(r,r')\ + ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

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adjoint operator algebra of projections Amer analytic arbitrary asymptotic B-space Banach space Boolean algebra Borel sets boundary conditions bounded Borel function bounded linear operator bounded operator bounded spectral commuting compact complex numbers complex plane constant contains continuous functions converges Corollary countably additive Definition denote differential operator disjoint Doklady Akad domain eigenvalues elements equation equivalent exists finite number Foias follows from Lemma follows from Theorem formal differential operator formula Hence Hilbert space hypothesis identity inequality inverse Lebesgue Math matrix multiplicity Nauk SSSR norm operators in Hilbert perturbation polynomial preceding Proof properties prove quasi-nilpotent restriction Russian satisfies scalar operator scalar type operator scalar type spectral Section semi-group sequence shows spectral measure spectral operator spectral theory spectrum strong operator topology subset Suppose trace class type spectral operator unbounded uniformly bounded unique vector weakly complete zero