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

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Page 2381

of ( 7 ) is necessarily unique for u € P + . Now define oilt , u ) to be the unique

2 qe ( cf. Corollary XIII.1.5 ) . If S is some other positive constant and S < , then ag

2 ...

of ( 7 ) is necessarily unique for u € P + . Now define oilt , u ) to be the unique

**solution**of ( T - u ? ) o = 0 , ( t , u ) € ( 0,00 ) X Pt , which coincides with gu ( t ) for t2 qe ( cf. Corollary XIII.1.5 ) . If S is some other positive constant and S < , then ag

2 ...

Page 2391

Appropriate choice of this second

properties of the resolvent as needed below . The following corollary summarizes

the necessary facts in a form convenient for later use . 6 COROLLARY . Let A ( a )

and ...

Appropriate choice of this second

**solution**will enable us to calculate finerproperties of the resolvent as needed below . The following corollary summarizes

the necessary facts in a form convenient for later use . 6 COROLLARY . Let A ( a )

and ...

Page 2394

Then there exists a

and for all sufficiently small ue P + , such that og and os are continuous in t and u

...

**solutions**ĝi = 0 ; lt , -u ( a ) ) and 01 = 01 ( t , u ( a ) ) of the equation to = λσ . ...Then there exists a

**solution**oz ( t , j ) of the equation to ru'o , defined for 0 St < ooand for all sufficiently small ue P + , such that og and os are continuous in t and u

...

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

47 other sections not shown

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adjoint operator Amer analytic apply arbitrary assumed B-space Banach space belongs Boolean algebra Borel set boundary conditions bounded bounded operator Chapter clear closed commuting compact complex constant contains continuous converges Corollary corresponding defined Definition denote dense determined differential operator discrete domain elements equation equivalent established exists extension fact finite follows formal formula function given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear operator Math Moreover multiplicity norm perturbation plane positive preceding present problem projections PROOF properties prove range resolution resolvent restriction Russian satisfies scalar type seen sequence shown shows similar solution spectral measure spectral operator spectrum subset sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector zero