## Linear Operators: Part III: Spectral Operators [by] Nelson Dunford and Jacob T. Schwartz, with the Assistance of William G. Bade and Robert G. Bartle, Volume 1 |

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Page 2197

Let E. - Va E , be the union of the increasing generalized

Let E. - Va E , be the union of the increasing generalized

**sequence**{ Ex } and let ε > 0 and X e X be arbitrary . Since E , X = sp EqX there is a vector y = ET - 1 Z ; and indices & ; such that Exq2 = 24 and Σ- = Eazy Ex - ) { a | E ...Page 2218

If a generalized

If a generalized

**sequence**of projections in a o - complete Boolean algebra of projections in a B - space converges weakly to a projection , then it converges strongly . PROOF . In view of Lemma 23 , the proof may be restricted to the ...Page 2450

By Theorem X.3.4 there exists a complete orthonormal basis { Xn } consisting of eigenvectors of T. We let Txn = dn Xn , and suppose that none of the eigenvalues 2n is multiple ( that is , that the

By Theorem X.3.4 there exists a complete orthonormal basis { Xn } consisting of eigenvectors of T. We let Txn = dn Xn , and suppose that none of the eigenvalues 2n is multiple ( that is , that the

**sequence**{ n } contains no repetitions ) ...### What people are saying - Write a review

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

31 other sections not shown

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adjoint operator Amer analytic apply arbitrary assumed B-space Banach space belongs Boolean algebra Borel set boundary conditions bounded bounded operator Chapter clear closed commuting compact complex constant contained continuous converges Corollary corresponding defined Definition denote dense determined differential operator domain elements equation equivalent established exists extension finite follows formal formula function given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator Math Moreover multiplicity norm 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 sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector zero