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

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Page 2051

We next

Cauchy problem whose existence is established in Theorem 19 . As before , we

assume that aạt > 0 . It follows from equation ( 55 ) which gives the analytical ...

We next

**consider**the problem of finding the explicit form of the solution to theCauchy problem whose existence is established in Theorem 19 . As before , we

assume that aạt > 0 . It follows from equation ( 55 ) which gives the analytical ...

Page 2409

That is , instead of

space L , ( D , X ) of all X - valued Borel - Lebesgue measurable functions defined

in D and satisfying lo ( 33 ) S 18 ( 2 , 4 ) o dx dy ) * = { S , 18 ( z ) dx dy } ip = 15 ...

That is , instead of

**considering**the space L , ( D ) as above , we may**consider**thespace L , ( D , X ) of all X - valued Borel - Lebesgue measurable functions defined

in D and satisfying lo ( 33 ) S 18 ( 2 , 4 ) o dx dy ) * = { S , 18 ( z ) dx dy } ip = 15 ...

Page 2488

( e ) Prove that if A is an operator of the Hilbert - Schmidt class , while we Lac ( H )

and | | w | ln < 00 , then * | Aet Hv12 dt $ 217 | | A | 13 | | | * · ( Hint :

adjoint , and expand in the eigenvectors of A . ) 17 ( a ) Let f be a continuous ...

( e ) Prove that if A is an operator of the Hilbert - Schmidt class , while we Lac ( H )

and | | w | ln < 00 , then * | Aet Hv12 dt $ 217 | | A | 13 | | | * · ( Hint :

**Consider**A selfadjoint , and expand in the eigenvectors of A . ) 17 ( a ) Let f be a continuous ...

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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