## 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 1951

Let T

Let T

**belong**to the right ( left ) ideal J in B ( X ) . Then every projection E ( o ) with 0 € ö**belongs**to J. If J is closed , then S and N also**belong**to J. PROOF . Let 0 € ō and let To = TE ( 0 ) | E ( 0 ) X , the restriction of T to ...Page 2264

then I

then I

**belongs**neither to the point nor to the residual spectrum of S. Using the formula for the spectral resolution of S given by the preceding theorem , we find that E ( { 2 } ) = 0 for #vo , deo ; thus , each such , must**belong**to ...Page 2462

Moreover , if C

Moreover , if C

**belongs**to the trace class 61 , then TnC converges to zero in trace norm , and CT * converges to zero in trace norm . Proof . The set K = C ( { x € H || * / $ 1 } ) is conditionally compact , and thus for each ε > 0 ...### What people are saying - Write a review

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

28 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 contains continuous converges Corollary corresponding countably additive defined Definition denote dense determined differential operator domain elements equation equivalent established exists extension fact finite follows formal formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit 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