## Linear operators: Spectral operators |

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Page 2307

The last example above illustrates a problem which will be of great concern to us

in the remainder of this chapter: the problem of finding which formal differential

operators and sets of

The last example above illustrates a problem which will be of great concern to us

in the remainder of this chapter: the problem of finding which formal differential

operators and sets of

**boundary conditions**lead to spectral operators. As our ...Page 2350

Spectral operators Nelson Dunford, Jacob T. Schwartz. V~1TU is the same as the

operator in Hilbert space defined by the formal differential operator (—idjdt + b'(t))

n and the set C,(£//) =0, i = 1 . . . n, of

Spectral operators Nelson Dunford, Jacob T. Schwartz. V~1TU is the same as the

operator in Hilbert space defined by the formal differential operator (—idjdt + b'(t))

n and the set C,(£//) =0, i = 1 . . . n, of

**boundary conditions**. Since ...Page 2371

Birkhoff [3 j showed that if the set of

regularity hypotheses of Section 4, the eigenvalue expansion of a function/ of

bounded variation converges to \{f(t + 0) +f(t — 0)} at an interior point t of the

interval [0, ...

Birkhoff [3 j showed that if the set of

**boundary conditions**is subject to theregularity hypotheses of Section 4, the eigenvalue expansion of a function/ of

bounded variation converges to \{f(t + 0) +f(t — 0)} at an interior point t of the

interval [0, ...

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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