## Linear operators: Spectral operators |

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Page 1939

A bounded operator T is a

follows from Lemma VII.3.4. Q.E.D. 4 Lemma. // S and N are bounded commuting

operators and if N is

corollary of ...

A bounded operator T is a

**quasi**-**nilpotent**if and only ifa(T)={0}. Proof. Thisfollows from Lemma VII.3.4. Q.E.D. 4 Lemma. // S and N are bounded commuting

operators and if N is

**quasi**-**nilpotent**, then a{S + N) = a(S). Proof. This is acorollary of ...

Page 2092

... we say that T and U are

\(U - T)[n]\Vn =0. (It is clear that if U = T + N where TN = NT is a

operator, then (T — U)M = {T — U)n, and T and U are

...

... we say that T and U are

**quasi**-**nilpotent**equivalent in case Urn \(T-U)M\lln=]im\(U - T)[n]\Vn =0. (It is clear that if U = T + N where TN = NT is a

**quasi**-**nilpotent**operator, then (T — U)M = {T — U)n, and T and U are

**quasi**-**nilpotent**equivalent;...

Page 2115

if T — U is

decomposable, then XT(F) = Xu(F) for all closed sets F if and only if T and U are ...

if T — U is

**quasi**-**nilpotent**.) It is proved that if T is decomposable and T and U are**quasi**-**nilpotent**equivalent, then U is decomposable. Moreover, if T and U aredecomposable, then XT(F) = Xu(F) for all closed sets F if and only if T and U are ...

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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 commuting compact complex numbers complex plane constant contains continuous functions converges Corollary countably additive Definition denote dense differential operator disjoint Doklady Akad domain eigenvalues elements equation exists finite number Foias follows from Lemma follows from Theorem formal differential operator formula Hence Hilbert space hypothesis identity inequality inverse Lebesgue Math matrix measurable functions multiplicity Nauk SSSR norm normal operators operators in Hilbert perturbation polynomial 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