## Linear Operators: Spectral Theory : Self Adjoint Operators in Hilbert Space, Volume 2 |

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Page 1926

EO ( T ) In terms of N and E the operational calculus for an

takes the form f ( T ) = " S fon ( A ) E ( NA ) , nro n ! JOT ) whereas if T is normal

only the first term f ( T ) = 1 . f ( ) E ( d ) O ( T ) of this series is needed to express f (

T ) .

EO ( T ) In terms of N and E the operational calculus for an

**arbitrary**operatortakes the form f ( T ) = " S fon ( A ) E ( NA ) , nro n ! JOT ) whereas if T is normal

only the first term f ( T ) = 1 . f ( ) E ( d ) O ( T ) of this series is needed to express f (

T ) .

Page 2101

This was proved by Dunford [ 18 ] ; however , McCarthy [ 1 , I ] showed that this

assertion does not hold in an

in an

...

This was proved by Dunford [ 18 ] ; however , McCarthy [ 1 , I ] showed that this

assertion does not hold in an

**arbitrary**B - space . In fact , all that can be inferredin an

**arbitrary**B - space is that Nm + 2 = 0 ; however , if o ( T ) has plane measure...

Page 2193

Q . E . D . The preceding corollary is not true if the Hilbert space is replaced by an

commuting scalar type spectral operators in a space of continuous functions ...

Q . E . D . The preceding corollary is not true if the Hilbert space is replaced by an

**arbitrary**B - space , for it has been shown by Kakutani [ 15 ] that the sum of twocommuting scalar type spectral operators in a space of continuous functions ...

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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 formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator manifold Math Moreover multiplicity norm normal 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