## Linear Operators: Spectral operators |

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Page 2257

point in o ( R ) , including zero , is contained in an arbitrarily small

point in o ( R ) , including zero , is contained in an arbitrarily small

**compact**set open in the relative topology of o ( R ) . Hence , o ( R ) is totally disconnected . It follows by Definition VII.9.3 that E ( o ; T ) = E ( + ( 0 ) ...Page 2357

then it is clear that L is a bounded operator and that ( S – XI ) -v = ( T - XI ) - L. Hence ( P + N ) ( S – XI ) -v = P ( S – XI ) - " + N ( S – 21 ) - " = P ( T - XI ) -L + N ( S - 1 ) - " is a bounded operator which is

then it is clear that L is a bounded operator and that ( S – XI ) -v = ( T - XI ) - L. Hence ( P + N ) ( S – XI ) -v = P ( S – XI ) - " + N ( S – 21 ) - " = P ( T - XI ) -L + N ( S - 1 ) - " is a bounded operator which is

**compact**if P ...Page 2462

is

is

**compact**. Put C = QR1 , and D = R2 , so that V = CD . The operator C is**compact**by Corollary VI.5.5 , and thus proof of Corollary 11 is complete . Q.E.D. 12 LEMMA . If C is a**compact**operator in H , and { Tn } is a uniformly bounded ...### What people are saying - Write a review

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### Contents

SPECTRAL OPERATORS 1937 1941 1945 XV Spectral Operators | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

32 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 function 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