## Linear Operators: Spectral operators |

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Page 2284

However , rather than seek the maximum generality , it will be convenient to

Both these properties hold for B if X is separable , so this will be assumed for the

rest of ...

However , rather than seek the maximum generality , it will be convenient to

**suppose**that B is itself complete and satisfies the countable chain condition .Both these properties hold for B if X is separable , so this will be assumed for the

rest of ...

Page 2303

enumeration of its spectrum . Let dn denote the distance from an to o ( T ) — { n } .

**Suppose**that E is its resolution of the identity , and**suppose**that { in } is anenumeration of its spectrum . Let dn denote the distance from an to o ( T ) — { n } .

**Suppose**that for all but a finite number of n , Elan ) has a onedimensional range .Page 2463

6 , we see that lim 810140 { I + ( i - d . ) ( 101 – H ) - 1 } * g = 0 for each ge H .

Statement ( b ) now follows from Lemma 12 . If | V ( 101 – H ) - 11 < 1 , then I – V (

101 – H ) - 1 has a bounded inverse by Lemma VII . 3 . 4 . Let us now

that I ...

6 , we see that lim 810140 { I + ( i - d . ) ( 101 – H ) - 1 } * g = 0 for each ge H .

Statement ( b ) now follows from Lemma 12 . If | V ( 101 – H ) - 11 < 1 , then I – V (

101 – H ) - 1 has a bounded inverse by Lemma VII . 3 . 4 . Let us now

**suppose**that I ...

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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