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

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Page 2189

Since E ( od ) = E ( 0 ) E ( S ) , it is also clear that s ( fg ) = S ( f ) S ( 9 ) if f and g

are characteristic functions of sets in E . For ... operator with the stated resolution

of the identity ,

8 ) ...

Since E ( od ) = E ( 0 ) E ( S ) , it is also clear that s ( fg ) = S ( f ) S ( 9 ) if f and g

are characteristic functions of sets in E . For ... operator with the stated resolution

of the identity ,

**let f**be in EB ( 1 , 2 ) and , for every Borel set d in the plane , let E (8 ) ...

Page 2248

Let T be a spectral operator , and E its resolution of identity .

analytic in a domain U which , when taken together with a finite number of

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

the point ...

Let T be a spectral operator , and E its resolution of identity .

**Let f**be a functionanalytic in a domain U which , when taken together with a finite number of

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

the point ...

Page 2488

( b )

derivative f ' is positive , continuous , and of bounded variation , and let V [ ( f ' ) –

1 ) be the total variation of the reciprocal of f ' . Prove that ab ell ( t ) dx < 2 max ( lf

...

( b )

**Let f**be a strictly increasing function on an interval [ a , b ] . Suppose that itsderivative f ' is positive , continuous , and of bounded variation , and let V [ ( f ' ) –

1 ) be the total variation of the reciprocal of f ' . Prove that ab ell ( t ) dx < 2 max ( lf

...

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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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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