## Linear Operators: Spectral operators |

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Page 1931

If x is a vector in X , then by an

X - valued function f defined and

that ( $ I – T ' ) f ( $ ) = x , & € D ( f ) . It is clear that , for such an extension , f ...

If x is a vector in X , then by an

**analytic**extension of R ( E ; T ' ) x will be meant anX - valued function f defined and

**analytic**on an open set D ( S ) z p ( T ) and suchthat ( $ I – T ' ) f ( $ ) = x , & € D ( f ) . It is clear that , for such an extension , f ...

Page 1932

In this case x ( K ) is a single valued

X ( $ ) = R ( É ; T ' ) x , Š Ep ( T ) . It will be shown in the next section that , if T is a

spectral operator , the function R ( F ; T ) x has , for every x in X , the single ...

In this case x ( K ) is a single valued

**analytic**function with domain p ( x ) and withX ( $ ) = R ( É ; T ' ) x , Š Ep ( T ) . It will be shown in the next section that , if T is a

spectral operator , the function R ( F ; T ) x has , for every x in X , the single ...

Page 2248

Let f be a function

number of exceptional points p , includes a neighborhood of o ( T ) and a

neighborhood of the point at infinity . Suppose that each exceptional point p

satisfies E ...

Let f be a function

**analytic**in a domain U which , when taken together with a finitenumber of exceptional points p , includes a neighborhood of o ( T ) and a

neighborhood of the point at infinity . Suppose that each exceptional point p

satisfies E ...

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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