## 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 2154

Since the

n + 1X + { x | ( T - XI ) n + 1x = 0 } is dense in X . By induction , it is seen that the

Since the

**manifolds**{ x | ( T – XI ) mx = 0 } increase with m , it follows that ( T – XI )n + 1X + { x | ( T - XI ) n + 1x = 0 } is dense in X . By induction , it is seen that the

**manifold**( T – 17 ) n + kX + { x | ( T – 27 ) n + Kx = 0 } is dense in X for all k 20 .Page 2214

To prove the converse , let A leave invariant every closed linear

invariant under every element of B and let B , be the strong closure of B . Then it

is clear that A leaves invariant every closed linear

To prove the converse , let A leave invariant every closed linear

**manifold**which isinvariant under every element of B and let B , be the strong closure of B . Then it

is clear that A leaves invariant every closed linear

**manifold**which is invariant ...Page 2270

In particular the

= { x * | ( I – E ) * 2 * = 0 } is X - closed is a standard elementary property of the

null

In particular the

**manifolds**E * * * , E * € B * , are X - closed . PROOF . That E * X *= { x * | ( I – E ) * 2 * = 0 } is X - closed is a standard elementary property of the

null

**manifold**of an adjoint operator . Let { E , } be an arbitrary set ( sequence ) of ...### What people are saying - Write a review

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### Contents

SPECTRAL OPERATORS XV Spectral Operators | 1924 |

Introduction | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

32 other sections not shown

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### Common terms and phrases

analytic apply arbitrary assumed B-space Banach space Boolean algebra Borel sets boundary conditions bounded bounded Borel bounded operator Chapter clear clearly closure commuting compact complex consider constant contained converges Corollary corresponding countably additive defined Definition denote dense determined differential operator domain elements equation equivalent established example exists extension fact finite follows formal formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator manifold Math Moreover multiplicity norm positive preceding present problem projections PROOF properties prove range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows spectral measure spectral operator spectrum statement strongly subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector weakly zero