## Linear Operators: Spectral operators |

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Page 2384

Q . E . D . For the spectral analysis of the operator T , we shall also need

asymptotic information on the “ second

u2o , that is , the

such a ...

Q . E . D . For the spectral analysis of the operator T , we shall also need

asymptotic information on the “ second

**solution**" of the differential equation To =u2o , that is , the

**solution**asymptotic to e - tut as t 00 . Since , in contrast to 01 ,such a ...

Page 2391

For use in what is to follow we record the formula for R ( a ; T ) obtained in the

proof of Lemma 4 and also observe that in constructing the kernel R ( s , t ; 2 ) in

Lemma 4 we may replace the particular “ second

in ...

For use in what is to follow we record the formula for R ( a ; T ) obtained in the

proof of Lemma 4 and also observe that in constructing the kernel R ( s , t ; 2 ) in

Lemma 4 we may replace the particular “ second

**solution**” oz ( t , d ) constructedin ...

Page 2394

On the other hand , since these two

have a linear relation 02 = aớı + 601 . ... Then there exists a

the equation to = u o , defined for 0 St < oo and for all sufficiently small u € P + ...

On the other hand , since these two

**solutions**are linearly independent , we musthave a linear relation 02 = aớı + 601 . ... Then there exists a

**solution**oz ( t , u ) ofthe equation to = u o , defined for 0 St < oo and for all sufficiently small u € P + ...

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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