## Linear Operators: Spectral operators |

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Page 2257

3 that Elo ; T ' ) = E ( + ( 0 ) ; R ) for each

is clear that as o runs over the family K of all

) runs over the family of all

3 that Elo ; T ' ) = E ( + ( 0 ) ; R ) for each

**compact**spectral set o of T . Moreover , itis clear that as o runs over the family K of all

**compact**open subsets of olT ' ) , 7 ( o) runs over the family of all

**compact**open subsets of o ( R ) which do not ...Page 2357

then it is clear that L is a bounded operator and that ( - XI ) - v = ( T – WI ) - " L .

Hence ( P + N ) ( S – XI ) - V = P ( S – XI ) - " + N ( S – XI ) - = P ( T – 2 ] ) - ' L + N (

S – AI ) - V is a bounded operator which is

cf .

then it is clear that L is a bounded operator and that ( - XI ) - v = ( T – WI ) - " L .

Hence ( P + N ) ( S – XI ) - V = P ( S – XI ) - " + N ( S – XI ) - = P ( T – 2 ] ) - ' L + N (

S – AI ) - V is a bounded operator which is

**compact**if P ( T – 21 ) - " is**compact**(cf .

Page 2462

is

. D . 12 LEMMA . If C is a

bounded ...

is

**compact**. Put C = QR1 , and D = Rg , so that V = CD . The operator C is**compact**by Corollary V1 . 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 uniformlybounded ...

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### Contents

SPECTRAL OPERATORS | 1924 |

An Operational Calculus for Bounded Spectral | 1941 |

Part | 1950 |

Copyright | |

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adjoint operator analytic apply arbitrary assume B-space Banach space Boolean algebra Borel sets boundary conditions bounded bounded operator Chapter clear closed commuting compact complex consider constant contained continuous converges Corollary corresponding countably additive defined Definition denote dense determined differential operator discrete 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 Math Moreover multiplicity norm positive preceding present problem projections PROOF properties proved range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows similar spectral measure spectral operator spectrum subset subspace sufficiently Suppose Theorem theory topology unbounded uniform uniformly unique valued vector weakly zero